# Arithmetics on number systems with irrational bases

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Arithmetics on number systems with irrational bases P. Ambrož C. Frougny Z. Masáková E. Pelantová Abstract For irrational β > 1 we consider the set Fin(β) of real numbers for which x has a finite number of non-zero digits in its expansion in base β. In particular, we consider the set of β-integers, i.e. numbers whose β-expansion is of the form n i=0 x i β i, n 0. We discuss some necessary and some sufficient conditions for Fin(β) to be a ring. We also describe methods to estimate the number of fractional digits that appear by addition or multiplication of β- integers. We apply these methods among others to the real solution β of x 3 = x 2 + x + 1, the so-called Tribonacci number. In this case we show that multiplication of arbitrary β-integers has a fractional part of length at most 5. We show an example of a β-integer x such that x x has the fractional part of length 4. By that we improve the bound provided by Messaoudi [11] from value 9 to 5; in the same time we refute the conjecture of Arnoux that 3 is the maximal number of fractional digits appearing in Tribonacci multiplication. 1 Introduction Let β be a fixed real number greater than 1 and let x be a positive real number. A convergent series n k= x k β k is called a β-representation of x if x = n k= x k β k and for all k the coefficient x k is a non-negative integer. If moreover for every < N < n we have N x k β k < β N+1 k= 1991 Mathematics Subject Classification :. Key words and phrases :. Bull. Belg. Math. Soc. 10(2003), 1 19

2 2 P. Ambrož C. Frougny Z. Masáková E. Pelantová the series n k= x k β k is called the β-expansion of x. The β-expansion is an analogue of the decimal or binary expansion of reals and we sometimes use the natural notation (x) β = x n x n 1 x 0 x 1 (n k)times Every x 0 has a unique β-expansion which is found by the greedy algorithm [13]. We can introduce lexicographic ordering on β-representations in the following way. The β-representation x n β n + x n 1 β n 1 + is lexicographically greater than x k β k + x k 1 β k 1 +, if k n and for the corresponding infinite words we have x n x n 1 00 } {{ 0} x k x k 1, where the symbol means the common lexicographic ordering on words in an ordered alphabet. Since the β-expansion is constructed by the greedy algorithm, it is the lexicographically greatest among all β-representations of x. The set of those x R, for which the β-expansion of x has only finitely many non-zero coefficients - digits, is denoted by Fin(β). Real numbers x with β-expansion of x of the form n k=0 x k β k are called β-integers and the set of β-integers is denoted by Z β. Let x Fin(β), x > 0 and let n k= N x k β k be its β-expansion with x N 0. If N > 0 then r = 1 k= N x kβ k is called the β-fractional part of x. If N 0 we set fp(x) = 0, for N > 0 we define fp(x) = N, i.e. fp(x) is the number of fractional digits in the β-expansion of x. Note that x is in Z β if and only if fp(x) = 0. If β Z, β > 1, then Fin(β) is closed under the operations of addition, subtraction and multiplication, i.e. Fin(β) is a ring. It is also easy to determine the β-expansion of x + y, x y, and x y with the knowledge of the β-expansions of x and y. In case that β > 1 is not a rational integer, the situation is more complicated and generally we don t know any criterion which would decide whether Fin(β) is a ring or not. Known results for β satisfying specific conditions will be mentioned below. Fin(β) being closed under addition implies it is closed under multiplication as well. Besides the question whether results of these arithmetic operations are always finite or not, we are also interested to describe the length of the resulting β-fractional part. It is possible to convert the sum x + y and the product x y of two numbers x, y Fin(β) by multiplication by a suitable factor β k into a sum or product of two β-integers. Therefore we define: L = L (β) := max {fp(x + y) x, y Z β, x + y Fin(β)}, L = L (β) := max {fp(x y) x, y Z β, x y Fin(β)}. The article is organized as follows. In Preliminaries we state some number theoretical facts and properties of Z β used below. The paper contains results of two types. Section 3 provides some necessary and some sufficient conditions on β in order that Fin(β) has a ring structure. The proofs of the sufficient conditions are constructive and thus provide an algorithm for addition of finite β-expansions. The second type of results gives estimates on the constants L (β), L (β). In Section 4 we explain a simple and effective method for determining these bounds. In Section 5 we apply this method to the case of the Tribonacci number, β the real solution of

3 Arithmetics on number systems with irrational bases 3 x 3 = x 2 + x + 1. We show 5 L (β) 6 and 4 L (β) 5 which improves the bound on L (β) = 9 given by Messaoudi [11] and refute the conjecture of Arnoux. The next Section 6 shows that the above method cannot be used for every β and therefore in Section 7 we introduce a different method that provides estimates for all Pisot numbers β. We illustrate its application on an example in Section 8. In Section 9 we mention some open problems related to the arithmetics on finite and infinite β-expansions. 2 Preliminaries The definitions recalled below and related results can be found in the survey [10, Chap. 7]. One can decide whether a particular β-representation of a number is also its β- expansion according to Parry s condition [12]. For fixed β > 1 we define a mapping T β : [0, 1] [0, 1) by the prescription T β (x) = βx [βx], where [z] is the greatest integer smaller or equal to z. The sequence d β (1) = t 1 t 2 t 3 such that t i = [βtβ i 1 (1)] is called the Rényi development of 1. Obviously, the numbers t i are non-negative integers smaller than β, t 1 = [β] and 1 = t i β i. (1) i=1 In order to state the Parry condition, we introduce the sequence d β(1) in the following way: If d β (1) has infinitely many non-zero digits t i, we set d β (1) = d β(1). If m is the greatest index of a non-zero digit in d β (1), i.e., 1 = t 1 β + t 2 β t m β m, t m 0, we set d β (1) = ( t 1 t 2 t m 1 (t m 1) )( t 1 t 2 t m 1 (t m 1) ), i.e., d β (1) is an infinite periodic sequence of period of length m. Theorem 2.1 (Parry). Let n k= x k β k be a β-representation of a number x > 0. Then n k= x k β k is the β-expansion of x if and only if for all i n the sequence x i x i 1 x i 2 is strictly lexicographically smaller than the sequence d β (1), symbolically x i x i 1 x i 2 d β(1).

4 4 P. Ambrož C. Frougny Z. Masáková E. Pelantová Notice that a finite string c n c n 1 c 0 of non-negative integers satisfies the Parry condition above, if c i c i 1 c 0 d β (1) for all i = 0, 1,..., n. Such strings are called admissible. A finite string of non-negative integers is called forbidden, if it is not admissible. Numbers β > 1 with eventually periodic Rényi development of 1 are called betanumbers [12]. It is easy to see that every beta-number is a solution of an equation x n a n 1 x n 1 a 1 x a 0 = 0, a n 1,...,a 1, a 0 Z. (2) Therefore β is an algebraic integer. If d is the minimal degree of a polynomial of the form (2), with root β, we say that d is the degree of β. The other roots β (2),...,β (d) of the polynomial are called the algebraic conjugates of β. As usually, we denote by Q(γ) the minimal subfield of complex numbers C which contains γ and the rationals Q. For an algebraic number β it is known that Q(β) = {g(β) g is a polynomial with coefficients in Q}. For 2 i d the mapping Q(β) Q(β (i) ) given by prescription z = g(β) z (i) = g(β (i) ) is an isomorphism of fields Q(β) and Q(β (i) ). In case that we consider only one resp. two conjugates of β we use instead of β (2), resp. β (3) the notation β, resp. β, and similarly we use z, resp. z for z (2), resp. z (3). Besides the eventually periodic Rényi development of 1, beta-numbers have another nice property: the set of distances between neighbors in Z β is finite and can be determined with the knowledge of d β (1). The distances have the values i = k=0 t k+i, for some i = 1, 2,..., βk+1 see [16]. From the definition of d β (1) it is obvious that the largest distance between neighboring β-integers is t k+1 1 = β = 1. k+1 k=0 It is not simple to give an algebraic description of the set of beta-numbers [15], however, it is known that every Pisot number is a beta-number [4]. Recall that Pisot numbers are those algebraic integers β > 1 whose all conjugates are in modulus less than 1. Other results about beta-numbers can be found in [5, 6].

6 6 P. Ambrož C. Frougny Z. Masáková E. Pelantová with β-expansions x = n k=0 a k β k, y = n k=0 b k β k. Then n k=0 (a k + b k )β k is a β- representation of the sum x + y. If the sequence of coefficients (a n + b n )(a n 1 + b n 1 ) (a 0 + b 0 ) verifies the Parry condition (Theorem 2.1), we have directly the β-expansion of x + y. In the opposite case, the sequence must contain a forbidden string. The so-called minimal forbidden strings defined below play a special role in our considerations. Definition 3.2. Let β > 1. A forbidden string u k u k 1 u 0 of non-negative integers is called minimal, if (i) u k 1 u 0 and u k u 1 are admissible, and (ii) u i 1 implies u k u i+1 (u i 1)u i 1 u 0 is admissible, for all i = 0, 1,..., k. Obviously, a minimal forbidden string u k u k 1 u 0 contains at least one non-zero digit, say u i 1. The forbidden string u k u k 1 u 0 with u i 1 is a β-representation of the addition of the two β-integers w = u k β k + +u i+1 β i+1 +(u i 1)β i +u i 1 β i u 0 and w = β i. The β-expansion of a number is lexicographically the greatest among all its β-representations, and thus if the sum w + w belongs to Fin(β), then there exists a finite β-representation of w + w lexicographically strictly greater than u k u k 1 u 0 (the β-expansion of w + w). We have thus shown a necessary condition on β in order that Fin(β) is closed under addition of two positive elements. The condition will be formulated as Proposition 3.4. For that we need the following definition. Definition 3.3. Let k, p Z, k p and z = z k β k + z k 1 β k z p β p, where z i N 0 for p i k. A finite β-representation of z of the form v n β n + v n 1 β n v l β l, l n, is called a transcription of z k β k + z k 1 β k z p β p if k n and v n v n 1 v l 00 } {{ 0} z k z p. (n k) times Proposition 3.4 (Property T). Let β > 1. If Fin(β) is closed under addition of positive elements, then β satisfies Property T: For every minimal forbidden string u k u k 1 u 0 there exists a transcription of u k β k + u k 1 β k u 1 β + u 0. If Property T is satisfied, the transcription of a β-representation of a number z can be obtained in the following way. Every β-representation of z which contains a forbidden string can be written as a sum of a minimal forbidden string β j (u k β k + + u 1 β + u 0 ) and a β-representation of some number z. The new transcribed β- representation of z is obtained by digit-wise addition of the transcription β j (v n β n + + v l β l ) of the minimal forbidden string and the β-representation of z. Example 3.5. Let us illustrate the notions we have introduced sofar on the simple example of the numeration system based on the golden ratio τ = 1 2 (1 + 5). Since τ satisfies x 2 = x+1, its Rényi development of 1 is d τ (1) = 11. The Parry condition implies that every finite word lexicographically greater or equal to 11 is forbidden.

7 Arithmetics on number systems with irrational bases 7 Thus admissible are only those finite words x k x l for which x i {0, 1} and x i x i 1 = 0, for l < i k. According to the definition, minimal forbidden words are 2 and 11. The golden ratio τ satisfies Property T, since 2 = τ + τ 2 and τ + 1 = τ 2, where and The rules we have derived can be used for obtaining the τ-expansion starting from any τ-representation of a given number x. For example 3 = = 1 + τ + 1 τ = 2 τ2 + 1 τ. 2 The τ-representations τ τ 2 and τ 2 + τ 2 are transcriptions of 3. In the same time is the τ-expansion of the number x = 3, since it does not contain any forbidden string. Let z k β k + z k 1 β k z p β p be a β-representation of z such that the string z k z k 1 z p is not admissible. Repeating the above described process we obtain a lexicographically increasing sequence of transcriptions. In general, it can happen that the procedure may be repeated infinitely many times without obtaining the lexicographically greatest β-representation of z, i.e. the β-expansion of z. The following theorems provide sufficient conditions in order that such situation is avoided. Theorem 3.6. Let β > 1. Suppose that for every minimal forbidden string u k u k 1 u 0 there exists a transcription v n β n +v n 1 β n 1 + +v l β l of u k β k +u k 1 β k u 1 β + u 0 such that v n + v n v l u k + u k u 0. Then Fin(β) is closed under addition of positive elements. Moreover, for every positive x, y Fin(β), the β-expansion of x+y can be obtained from any β-representation of x + y using finitely many transcriptions. Proof. Without loss of generality, it suffices to decide about finiteness of the sum x + y where x = n k=0 a k β k and y = n k=0 b k β k are the β-expansions of x and y, respectively. We prove the theorem by contradiction, i.e., suppose that we can apply a transcription to the β-representation n k=0 (a k + b k )β k infinitely many times. We find M N such that x + y < β M+1. Then the β-representation of x + y obtained after the k-th transcription is of the form x + y = M i=l k c (k) i β i, where l k is the smallest index of non-zero coefficient in the β-representation in the k-th step. Since for every exponent i Z there exists a non-negative integer f i such that x + y f i β i, we have that c (k) i f i for every step k. Realize that for every index p Z, p M, there are only finitely many sequences c M c M 1 c p satisfying 0 c i f i for all i = M, M 1,..., p. Since in every step k the sequence c (k) M c (k) M 1 lexicographically increases, we can find for every index p the step κ, so that the digits c (k) M, c (k) we have M 1,..., c (k) are constant for k κ. Formally, ( p Z, p M)( κ N)( k N, k κ)( i Z, M i p)(c (k) i p = c (κ) i ) (5)

8 8 P. Ambrož C. Frougny Z. Masáková E. Pelantová Since by assumption of the proof, the transcription can be performed infinitely many times, it is not possible that the digits c (κ) i for i < p are all equal to 0. Let us denote by t the maximal index t < p with non-zero digit, i.e., c κ t 1. In order to obtain the contradiction, we use the above idea (5) repeatedly. For p = 0 we find κ 1 and t 1 satisfying x + y = M i=0 c (κ 1) i β i + c (κ 1) t 1 β t 1 + t 1 1 i=l κ1 c (κ1) i β i. In further steps k κ 1 the digit sum M i=0 c (k) i remains constant, since the digits c (k) i remain constant. The digit sum ( 1) i=t 1 c (k) i 1, because the sequence of digits lexicographically increases. For every k κ 1 we therefore have M i=t 1 c (k) i 1 + M i=0 c (k) i. We repeat the same considerations for p = t 1. Again, we find the step κ 2 > κ 1 and the position t 2 < t 1, so that for every k κ 2 M i=t 2 c (k) i 1 + M c (k) i. i=t 1 In the same way we apply (5) and find steps κ 3 < κ 4 < κ 5 <... and positions t 3 > t 4 > t 5 such that the digit sum M i=ts c (κs) i increases with s at least by 1. Since there are infinitely many steps, the digit sum increases with s to infinity, which contradicts the fact that we started with the digit sum n k=0 (a k +b k ) and the transcription we use do not increase the digit sum. Let us comment on the consequences of the proof for β satisfying the assumptions of the theorem. The β-expansion of the sum of two β-integers can be obtained by finitely many transcriptions where the order in which we transcribe the forbidden strings in the β-representation of x + y is not important. However, the proof does not provide an estimate on the number of steps needed. Recall that for rational integers the number of steps depends only on the number of digits of the summated numbers. It is an interesting open problem to determine the complexity of the summation algorithm for β-integers. In order to check whether β satisfies Property T we have to know all the minimal forbidden strings. Let the Rényi development of 1 be finite, i.e., d β (1) = t 1 t 2 t m. Then a minimal forbidden string has one of the forms (t 1 + 1), t 1 (t 2 + 1), t 1 t 2 (t 3 + 1),..., t 1 t 2 t m 2 (t m 1 + 1), t 1 t 2 t m 1 t m. Note that not all the above forbidden strings must be minimal. For example if β has the Rényi development d β (1) = 111, the above list of strings is equal to 2, 12, 111. However, 12 is not minimal. In [8] it is shown that if β has a finite development of 1 with decreasing digits, then Fin(β) is closed under addition. The proof includes an algorithm for addition. Let us show that the result of [8] is a consequence of our Theorem 3.6.

9 Arithmetics on number systems with irrational bases 9 Corollary 3.7. Let d β (1) = t 1 t m, t 1 t 2 t m 1. Then Fin(β) is closed under addition of positive elements. Proof. We shall verify the assumptions of Theorem 3.6. Consider the forbidden string t 1 t 2 t i 1 (t i + 1), for 1 i m 1. Clearly, the following equality is verified t 1 β i t i 2 β 2 + t i 1 β + (t i + 1) = = β i + (t 1 t i+1 )β (t m i t m )β m+i + t m i+1 β m+i t m β m. The assumption of the corollary assure that the coefficients on the right hand side are non-negative. The digit sum on the left and on the right is the same. Thus 1 } 0 0 {{ 0} (t 1 t i+1 ) (t 2 t i+2 ) (t m i t m ) i times is the desired finite string lexicographically strictly greater than 0 t 1 t 2 t i 1 (t i +1). It remains to transcribe the string t 1 t 2 t m 1 t m into the lexicographically greater string 1 } 0 0 {{ 0}. m times The conditions of Theorem 3.6 are however satisfied also for other irrationals that do not fulfil assumptions of Corollary 3.7. As an example we may consider the minimal Pisot number. It is known that the smallest among all Pisot numbers is the real solution β of the equation x 3 = x + 1. The Rényi development of 1 is d β (1) = The number β thus satisfies relations β 3 = β + 1 and β 5 = β The minimal forbidden strings are 2, 11, 101, 1001, and Their transcription according to Property T is the following: 2 = β 2 + β 5 β + 1 = β 3 β = β 3 + β 3 β = β 4 + β 5 β = β 5 The digit sum in every transcription is smaller than or equal to the digit sum of the corresponding minimal forbidden string. Therefore Fin(β) is according to Theorem 3.6 closed under addition of positive numbers. Since d β (1) is finite, by Proposition 3.1 Fin(β) is a ring. This was shown already in [2]. In the assumptions of Theorem 3.6 the condition of non-increasing digit sum can be replaced by another requirement. We state it in the following theorem. Its proof uses the idea and notation of the proof of Theorem 3.6. Theorem 3.8. Let β > 1 be an algebraic integer satisfying Property T, and let at least one of its conjugates, say β, belong to (0, 1). Then Fin(β) is closed under addition of positive elements. Moreover, for every positive x, y Fin(β), the β- expansion of x + y can be obtained from any β-representation of x + y using finitely many transcriptions.

10 10 P. Ambrož C. Frougny Z. Masáková E. Pelantová Proof. If it was possible to apply a transcription on the β-representation of x + y infinitely many times, then we obtain the sequence of β-representations x + y = M i=l k c (k) i β i, where the smallest indices of the non-zero digits l k satisfy lim k l k =. Here we have used the notation of the proof of Theorem 3.6. Now we use the isomorphism between algebraic fields Q(β) and Q(β ) to obtain (x + y) = x + y = M i=l k c (k) i (β ) i (β ) l k. The last inequality follows from the fact that β > 0 and c (k) i 0 for all k and i. Since β < 1 we have lim k (β ) l k = +, which is a contradiction. Remark 3.9. Let us point out that an algebraic integer β with at least one conjugate in the interval (0, 1) must have an infinite Rényi development of 1. Such β has necessarily infinitely many minimal forbidden strings. The only examples known to the authors of β satisfying Property T and having a conjugate β (0, 1) have been treated in [8], namely those which have eventually periodic d β (1) with period of length 1, d β (1) = t 1 t 2 t m 1 t m t m t m, with t 1 t 2 t m 1. (6) In such a case every minimal forbidden string has a transcription with digit sum strictly smaller than its own digit sum. Thus closure of Fin(β) under addition of positive elements follows already by Theorem 3.6. This means that we don t know any β for which Theorem 3.8 would be necessary. From the above remark one could expect that the fact that Fin(β) is closed under addition forces the digit sum of the transcriptions of minimal forbidden strings to be smaller than or equal to the digit sum of the corresponding forbidden string. It is not so. For example let β be the solution of x 3 = 2x Then d β (1) = 201 and the minimal forbidden string 3 has the β-expansion 3 = β + 1 β + 1 β β β 4. The digit sum of this transcription of 3 is equal to 4. If there exists another transcription of 3 with digit sum 3, it must be lexicographically strictly larger than 03 and strictly smaller than , because the β-expansion of a number is lexicographically the greatest among all its β-representations. It can be shown easily that a string with the above properties does not exist. In the same time Fin(β) is closed under addition. This follows from the results of Akiyama who shows that for a cubic Pisot unit β the set Fin(β) is a ring if and only if d β (1) is finite, see [2]. We can also use the result of Hollander [9] who shows that Fin(β) is a ring for β > 1 root of the equation x m = a m 1 x m 1 + +a 1 x+a 0, such that a m 1 > a m 2 + +a 1 +a 0 a i N.

11 Arithmetics on number systems with irrational bases 11 On the other hand, Property T is not sufficient for Fin(β) to be closed under addition of positive elements. As an example we can mention β with the Rényi development of 1 being d β (1) = Such β satisfies β 6 = β Among the conjugates of β there is a pair of complex conjugates, say β, β = β, with absolute value β = β Thus β is not a Pisot number and according to the result of [8], Fin(β) cannot be closed under addition of positive elements. However, Property T is satisfied for β. All minimal forbidden string can be transcribed as follows: 2 = β + β 6 + β 7 + β 8 + β 9 + β 10 β + 1 = β 2 + β 6 + β 7 + β 8 + β 9 β = β 3 + β 6 + β 7 + β 8 β = β 4 + β 6 + β 7 β = β 5 + β 6 β = β 6 The expressions on the right hand side are the desired transcriptions, since they are finite and lexicographically strictly greater than the corresponding minimal forbidden string. 4 Upper bounds on L and L As we have mentioned in the beginning of the previous section, Fin(β) is a ring only for β a Pisot number. However, it is meaningful to study upper bounds on the number of fractional digits that appear as a result of addition and multiplication of β-integers also in the case that Fin(β) is not a ring. We shall explain two methods for determining upper estimates on L (β) and L (β). The first method is applicable even in the case that β is not a Pisot number. The first method stems from the theorem that we cite from [7]. The idea of the theorem is quite simple and was in some way used already by other authors [11, 17]. It uses the isomorphism between the fields Q(β) and Q(β ) which to a β-integer z = n i=0 c i β i assigns its algebraic conjugate z = n i=0 c i β i. Theorem 4.1. Let β be an algebraic number, β > 1, with at least one conjugate β satisfying H := sup{ z z Z β } < +, K := inf{ z z Z β \ βz β } > 0. Then ( ) L 1 < 2H β K Remark 4.2. and ( ) L 1 < H2 β K. (7) Since H sup{ β k k N}, the condition H < + implies that β < 1. In this case H [β] β i = [β] 1 β. i=0

12 12 P. Ambrož C. Frougny Z. Masáková E. Pelantová If β (0, 1), we have for z Z β \ βz β that z = n i=0 c i β i c 0 1. The value 1 is achieved for z = 1. Therefore K = 1. If the considered algebraic conjugate β of β is negative or complex, it is complicated to determine the value of K and H. However, for obtaining bounds on L, L it suffices to have reasonable estimates on K and H. In order to determine good approximation of K and H we introduce some notation. For n N we shall consider the set E n := {z Z β 0 z < β n }. In fact this is the set of all a 0 + a 1 β + + a n 1 β n 1 where a n 1 a 1 a 0 is an admissible β-expansion. We denote min n := min{ z z E n, z / βz β }, max n := max{ z z E n }. Lemma 4.3. Let β > 1 be an algebraic number with at least one conjugate β < 1. Then (i) For all n N we have K K n := min n β n H. (ii) K > 0 if and only if there exists n N such that K n > 0. Proof. (i) Let z Z β \ βz β. Then the β-expansion of z is z = N i=0 b i β i, b 0 0. The triangle inequality gives n 1 z b i β i N b i β i min n β n N b i β i n > min n β n H = K n. i=0 i=n Hence taking the infimum on both sides we obtain K K n. (ii) From the definition of min n it follows that min n is a decreasing sequence with lim n min n = K. If there exists n N such that min n β n H > 0 we have K > 0 from (i). The opposite implication follows easily from the fact that lim n K n = K. For a fixed β, the determination of min n for small n is relatively easy. It suffices to find the minimum of a finite set with small number of elements. If for such n we have K n = min n β n H > 0, we obtain using (7) bounds on L and L. We illustrate this procedure on the real solution β of x 3 = x 2 + x + 1, the so-called Tribonacci number. i=n 5 L, L for the Tribonacci number Let β be the real root of x 3 = x 2 + x + 1. The arithmetics on β-expansions was already studied in [11]. Messaoudi [11] finds the upper bound on the number of β- fractional digits for the Tribonacci multiplication as 9. Arnoux, see [11], conjectures that L = 3. We refute the conjecture of Arnoux and find a better bound on L than 9. Moreover, we find the bound for L, as well.

13 Arithmetics on number systems with irrational bases 13 It turns out that the best estimates on L, L are obtained by Theorem 4.1 with approximation of K by K n for n = 9. By inspection of the set E 9 we obtain and min 9 = 1 + β 2 + β 4 + β max 9 = 1 + β 3 + β Consider y Z β, y = N k=0 a k β k. Then from the triangle inequality y 8 a k β k + β 9 17 a k β k 9 + β a k β k 18 + k=0 k=9 k=18 ( < max β 9 + β 18 + ) = max 9 1 β. 9 In this way we have obtained an upper estimate on H, i.e., H max 9 1 β 9. This implies Hence K 9 = min 9 β 9 H min 9 β 9 max 9 1 β 9. ( ) L 1 < 2H β K 2 max ( ) 1 9 min 1 β 9 9 β 9 max β 9 ( ) L ( ) 2 ( ) 1 1 < H2 β K max9 min 1 β 9 9 β 9 max β 9 Since ( ) , β ( ) , β ( ) , β we conclude that L 6, L 5. In order to determine the lower bounds on L, L we have used a computer program [3] to perform additions and multiplications on a large set of β-expansions. As a result we have obtained examples of a sum with 5 and product with 4 fractional digits, namely: = = We can thus sum up our results as 5 L 6 and 4 L 5.

14 14 P. Ambrož C. Frougny Z. Masáková E. Pelantová 6 Case K = 0 The above mentioned method cannot be used in case that K = 0. It is however difficult to prove that K = 0 for a given algebraic β and its conjugate β. Particular situation is solved by the following proposition. Proposition 6.1. Let β > 1 be an algebraic number and β ( 1, 0) its conjugate such that 1 β 2 < [β]. Then K = 0. Proof. Set γ := β 2. Digits in the γ-expansion take values in the set {0, 1,..., [γ]}. Since [γ] [β] 1 and the Rényi development of unit d β (1) is of the form d β (1) = [β]t 2 t 3, every sequence of digits in {0, 1,..., [γ]} is lexicographically smaller than d β (1) and thus is an admissible β-expansion. Since 1 < β 1 < γ, the γ-expansion of β 1 has the form β 1 = c 0 + c 1 γ 1 + c 2 γ 2 + c 3 γ 3 + (8) where all coefficients c i [β] 1. Let us define the sequence z n := 1 + c 0 β + c 1 β 3 + c 2 β c n β 2n+1. Clearly, z n Z β \ βz β and z n := 1 + β (c 0 + c 1 β 2 + c 2 β c n β 2n ). According to (8) we have lim n z n = 0 = lim n z n. Finally, this implies K = 0. An example of an algebraic number satisfying assumptions of Propositions 6.1 is β > 1 solution of the equation x 3 = 25x x + 2. The algebraic conjugates of β are β and β , and so K = 0 for both of them. Hence Theorem 4.1 cannot be used for determining the bounds on L, L. We thus present another method for finding these bounds and illustrate it further on the mentioned example. Note that similar situation happens infinitely many times, for example for a class of totally real cubic numbers, solutions of x 3 = p 6 x 2 +p 4 x+p, for p 3. Theorem 4.1 cannot be applied to any of them which justifies utility of a new method. 7 Upper bounds on L, L for β Pisot numbers The second method of determining upper bounds on L, L studied in this paper is applicable to β being a Pisot number, i.e., an algebraic integer β > 1 with conjugates in modulus less than 1. This method is based on the so-called cut-and-project scheme. Let β > 1 be an algebraic integer of degree d, let β (2),...,β (s) be its real conjugates and let β (s+1), β (s+2) = β (s+1),..., β (d 1), β (d) = β (d 1) be its non-real conjugates. Then there exists a basis x 1, x 2,..., x d of the space R d such that every x = (a 0, a 1,...,a d 1 ) Z d has in this basis the form x = α 1 x 1 + α 2 x α d x d, where α 1 = a 0 + a 1 β + a 2 β a d 1 β d 1 =: z Q(β)

15 Arithmetics on number systems with irrational bases 15 and α i = z (i) for i = 2, 3,..., s, α j = R(z (j) ) for s < j d, j odd, α j = I(z (j) ) for s < j d, j even. Technical details of the construction of the basis x 1, x 2,..., x d can be found in [1, 7, 17]. Let us denote Z[β] := {a 0 + a 1 β + a 2 β a d 1 β d 1 a i Z}. For β an algebraic integer, the set Z[β] is a ring and moreover it can be geometrically interpreted as a projection of the lattice Z d on a suitable chosen straight line in R d. The correspondence (a 0, a 1,..., a d 1 ) a 0 +a 1 β+a 2 β 2 + +a d 1 β d 1 is a bijection of the lattice Z d on the ring Z[β]. In the following, we shall consider β an irrational Pisot number. Important property that will be used is the inclusion Z β Z[β]. (9) Let us recall that Z β is a proper subset of Z[β], since Z[β] is dense in R as a projection of the lattice Z d, whereas Z β has no accumulation points. Since Z[β] is a ring, Z β + Z β Z[β] and Z β Z β Z[β]. Consider an x Z β with the β-expansion x = n k=0 a k β k. Then x (i) n = a k (β (i) ) k < [β] β (i) k = k=0 k=0 for every i = 2, 3,..., d. Therefore we can define The inclusion (9) thus can be precised to [β] 1 β (i), H i := sup{ x (i) x Z β } (10) Z β {x Z[β] x (i) < H i, i = 2, 3,..., d}. Another important property needed for determining upper bounds of L, L is the finiteness of the set C(l 1, l 2,..., l d ) := {x Z[β] x < l 1, x (i) < l i, i = 2, 3,..., d}, for every choice of positive l 1, l 2,...,l d. A point a 0 + a 1 β + + a d 1 β d 1 belongs to C(l 1, l 2,...,l d ) only if the point (a 0, a 1,..., a d 1 ) of the lattice Z d has all coordinates in the basis x 1,..., x d in a bounded interval ( l i, l i ), i.e., (a 0, a 1,...,a d 1 ) belongs to a centrally symmetric parallelepiped. Every parallelepiped contains only finitely many lattice points. Let us mention that notation C(l 1, l 2,...,l d ) is kept in accordance with [1], where Akiyama finds some conditions for Fin(β) to be a ring according to the properties of C(l 1, l 2,..., l d ). Our aim here is to use the set for determining the bounds on the length of the β-fractional part of the results of additions and multiplications in Z β.

16 16 P. Ambrož C. Frougny Z. Masáková E. Pelantová Theorem 7.1. Let β be a Pisot number of degree d, and let H 2, H 3,..., H d be defined by (10). Then L max{fp(r) r Fin(β) C(1, 2H 2, 2H 3,...,2H d )}, L max{fp(r) r Fin(β) C(1, H H 2,...,H 2 d + H d )}. Proof. Consider x, y Z β such that x + y > 0, x + y Fin(β). Set z := max{w Z β w x + y}. Then r := x + y z is the β-fractional part of x + y and thus r Fin(β) and fp(r) = fp(x+y), and 0 r < 1. Numbers x, y, z belong to the ring Z[β] and hence also r Z[β]. From the triangle inequality r (i) = x (i) + y (i) z (i) 2H i for all i = 2, 3,..., d. Therefore r belongs to the finite set C(1, 2H 2, 2H 3,...,2H d ), which together with the definition of L gives the statement of the theorem for addition. The upper bound on L is obtained analogically. 8 Application to β solution of x 3 = 25x x + 2. We apply the above Theorem 7.1 on β > 1 solution of the equation x 3 = 25x 2 +15x+ 2. Recall that such β satisfies the conditions of Proposition 6.1 for both conjugates β, β and thus Theorem 4.1 cannot be used for determining the bounds on L, L. Since [β] = 25, β-expansions are words in the 26-letter alphabet, say {(0), (1),..., (25)}. The Rényi development of 1 is d β (1) = (25)(15)(2). Since the digits of d β (1) are decreasing, Corollary 3.7 implies that the set Fin(β) is a ring. In case that some of the algebraic conjugates of β is a real number, the bounds from Theorem 7.1 can be refined. In our case β is totally real. Let x Z β, x = ni=0 a i β i. Since β < 0, we have n n x = a i (β ) i i=0 i=0,i even a i(β ) i < (25)(β ) 2i = 25 i=0 1 β 2 = H 1. The lower bound on x is Similarly for x we obtain n x = a i (β ) i n a i (β ) i > β H 1. i=0 i=0,i odd β H 2 < x < 25 1 β 2 = H 2. Consider x, y Z β such that x + y > 0. Again, the β-fractional part of x + y has the form r = x + y z for some z Z β. Thus (2β 1)H 1 = β H 1 + β H 1 H 1 < r = x + y z < H 1 + H 1 β H 1 = (2 β )H 1 (2β 1)H 2 < r = x + y z < (2 β )H 2

17 Arithmetics on number systems with irrational bases 17 We have used a computer to calculate explicitly the set of remainders r = A+Bβ + Cβ 2, A, B, C Z, satisfying 0 < A + Bβ + Cβ 2 < 1 (2β 1)H 1 < A + Bβ + Cβ 2 < (2 β )H 1 (2β 1)H 2 < A + Bβ + Cβ 2 < (2 β )H 2 where for β, β we use numerical values, (see Section 6). The set has 93 elements, which we shall not list here. For every element of the set we have found the corresponding β-expansion. The maximal length of the β-fractional part is 5. Thus L 5. On the other hand, using the algorithm described in Section 3 we have found a concrete example of addition of numbers (x) β = (25)(0)(25) and (y) β = (25)(0)(25), so that x + y has the β-expansion (x + y) β = (1)(24)(12)(11) (23)(0)(14)(13)(2). Thus we have found the value L = 5. In order to obtain bounds on L, we have computed the list of all r = A+Bβ + Cβ 2, A, B, C Z, satisfying the inequalities 0 < A + Bβ + Cβ 2 < 1 β H1 2 H 1 < A + Bβ + Cβ 2 < H1 2 β H 1 β H2 2 H 2 < A + Bβ + Cβ 2 < H2 2 β H 2 In this case we have obtained 8451 candidates on the β-fractional part of multiplication. The longest of them has 7 digits. From the other hand we have for (x) β = (25) and (y) β = (25), (x y) β = (24)(10) (21)(24)(16)(7)(16)(13)(2). Therefore L = 7. Let us mention that the above method can be applied also to the case of the Tribonacci number, but the bounds obtained in this way are not better than those from Theorem 4.1. We get L 6, L 6. 9 Comments and open problems 1. It is clear from the second method of estimate that for β a Pisot number L, L are finite numbers. Does there exist a β non Pisot such that L = + or L = +? 2. Whenever β satisfies Property T, it is possible to apply repeatedly the transcription on the β-representation of x + y, x, y Fin(β), x, y > 0. If the transcription can be applied infinitely many times, what is the order of choice of forbidden strings so that the sequence of β-representations converges rapidly to the β-expansion of x + y? 3. It is known [4, 14] that for β a Pisot number every x Q(β) has a finite or eventually periodic β-expansion. It would be interesting to have algorithms working with periodic expansions.

18 18 P. Ambrož C. Frougny Z. Masáková E. Pelantová Acknowledgements The authors are grateful for useful suggestions of the referees. We also thank Prof. S. Akiyama for pointing out some improvements in Theorem 7.1. The authors acknowledge financial support by the Grant Agency of the Czech Republic GA 201/01/0130; by the Barrande Project number 03900WJ, Z.M. and E.P. are grateful for the hospitality of LIAFA, Université Paris 7, and Ch.F. thanks the Department of Mathematics, FNSPE Czech Technical University, where part of the work was done. References [1] S. Akiyama, Self affine tiling and Pisot numeration system, in Number Theory and its Applications, Kyoto 1997, Eds. K. Györy and S. Kanemitsu, Kluwer Acad. Publ., Dordrecht (1999), [2] S. Akiyama, Cubic Pisot Units with finite beta expansions, in Algebraic Number Theory and Diophantine Analysis, Graz 1998, Eds. F.Halter-Koch and R.F. Tichy, de Gruyter, Berlin, (2000), [3] P. Ambrož, Computer program performing arithmetical operations on β- integers for a family of Pisot numbers: [4] A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sc. Paris, Série A, t.285, (1977), [5] D. W. Boyd, On the beta expansions for Salem numbers of degree 6, Math. Comp. 65 (1996) [6] D. W. Boyd, Salem numbers of degree four have periodic expansions, Théorie des nombres (Québec, PQ 1987), 57 64, de Gruyter, Berlin-New York, (1989). [7] L.S. Guimond, Z. Masáková, E. Pelantová, Arithmetics on β-expansions, to appear in Acta Arith. (2003). [8] Ch. Frougny and B. Solomyak, Finite β-expansions, Ergodic Theory Dynamical Systems 12 (1994), [9] M. Hollander, Linear numeration systems, finite beta-expansions, and discrete spectrum of substitution dynamical systems, Ph.D. Thesis, Washington University (1996). [10] M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, [11] A. Messaoudi, Généralisation de la multiplication de Fibonacci, Math. Slovaca 50 (2000),

19 Arithmetics on number systems with irrational bases 19 [12] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung. 11 (1960), [13] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957), [14] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), [15] B. Solomyak, Conjugates of beta-numbers and the zero-free domain for a class of analytic functions, Proc. London Math. Soc. 68 (1994), [16] W.P. Thurston, Groups, tilings, and finite state automata, Geometry supercomputer project research report GCG1, University of Minnesota (1989). [17] J.-L. Verger-Gaugry, J. P. Gazeau, Geometric study of the set of beta-integers for a Perron number and mathematical quasicrystals, to appear in J. Théor. Nombres Bordeaux (2003). Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University Trojanova 13, Praha 2, Czech Republic s: LIAFA, CNRS UMR place Jussieu, Paris Cedex 05, France and Université Paris 8

### s-convexity, model sets and their relation

s-convexity, model sets and their relation Zuzana Masáková Jiří Patera Edita Pelantová CRM-2639 November 1999 Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical

### Parallel Addition in Non-standard Numeration Systems

Parallel Addition in Non-standard Numeration Systems Christiane Frougny*, Edita Pelantová**, Milena Svobodová** * LIAFA, UMR 7089 CNRS, Université Paris 7 & Université Paris 8, Paris, France ** Dept. of

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Parallel Addition in Non-standard Numeration Systems

1 / 23 Parallel Addition in Non-standard Numeration Systems Christiane Frougny*, Edita Pelantová**, Milena Svobodová** * LIAFA, UMR 7089 CNRS, Université Paris 7 & Université Paris 8, Paris, France **

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### Continued fractions and good approximations.

Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

### Quadratic irrationalities and geometric properties of one-dimensional quasicrystals

Quadratic irrationalities and geometric properties of one-dimensional quasicrystals Zuzana Masáková Jiří Patera Edita Pelantová CRM-2565 September 1998 Department of Mathematics, Faculty of Nuclear Science

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### Lecture Notes on Polynomials

Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### 1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

### God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)

Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### Lecture 1: Elementary Number Theory

Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Generalized golden ratios of ternary alphabets

Generalized golden ratios of ternary alphabets V. Komornik Strasbourg Numération University of Liège June 8, 2011 V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, 2011 1 / 19 Abstract

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.

MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM

Acta Math. Univ. Comenianae Vol. LXXXI, (01), pp. 03 09 03 ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### A Curious Case of a Continued Fraction

A Curious Case of a Continued Fraction Abigail Faron Advisor: Dr. David Garth May 30, 204 Introduction The primary purpose of this paper is to combine the studies of continued fractions, the Fibonacci

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

### New Methods Providing High Degree Polynomials with Small Mahler Measure

New Methods Providing High Degree Polynomials with Small Mahler Measure G. Rhin and J.-M. Sac-Épée CONTENTS 1. Introduction 2. A Statistical Method 3. A Minimization Method 4. Conclusion and Prospects

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Appendix A: Numbers, Inequalities, and Absolute Values. Outline

Appendix A: Numbers, Inequalities, and Absolute Values Tom Lewis Fall Semester 2015 Outline Types of numbers Notation for intervals Inequalities Absolute value A hierarchy of numbers Whole numbers 1, 2,

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

### Approximating the entropy of a 2-dimensional shift of finite type

Approximating the entropy of a -dimensional shift of finite type Tirasan Khandhawit c 4 July 006 Abstract. In this paper, we extend the method used to compute entropy of -dimensional subshift and the technique

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### On the greatest and least prime factors of n! + 1, II

Publ. Math. Debrecen Manuscript May 7, 004 On the greatest and least prime factors of n! + 1, II By C.L. Stewart In memory of Béla Brindza Abstract. Let ε be a positive real number. We prove that 145 1

### FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS

FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### Solving a family of quartic Thue inequalities using continued fractions

Solving a family of quartic Thue inequalities using continued fractions Andrej Dujella, Bernadin Ibrahimpašić and Borka Jadrijević Abstract In this paper we find all primitive solutions of the Thue inequality

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

### Algebraic and Transcendental Numbers

Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### Chapter 6. Number Theory. 6.1 The Division Algorithm

Chapter 6 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS

1 CHAPTER 6. MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 Introduction Recurrence

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### Algebra I Notes Review Real Numbers and Closure Unit 00a

Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition,

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### How many numbers there are?

How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural

### Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)

Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

### ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES.

ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES. O. N. KARPENKOV Introduction. A series of properties for ordinary continued fractions possesses multidimensional

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Our Primitive Roots. Chris Lyons

Our Primitive Roots Chris Lyons Abstract When n is not divisible by 2 or 5, the decimal expansion of the number /n is an infinite repetition of some finite sequence of r digits. For instance, when n =

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### Gröbner Bases and their Applications

Gröbner Bases and their Applications Kaitlyn Moran July 30, 2008 1 Introduction We know from the Hilbert Basis Theorem that any ideal in a polynomial ring over a field is finitely generated [3]. However,

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28