Generalized golden ratios of ternary alphabets

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1 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, / 19

2 Abstract We report on a joint work with Anna Chiara Lai and Marco Pedicini on unique expansions in noninteger bases on three-letter alphabets: Generalized golden ratios of ternary alphabets, J. Eur. Math. Soc. 13 (2011), V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

3 EXPANSIONS We consider an alphabet {a 1 < < a J } of finitely many real digits; a real base q > 1. By an expansion of a real number x we mean a sequence c = (c i ) A satisfying c i x = q i. If x has a unique expansion, then its expansion is called a univoque sequence. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

4 EXAMPLES OF UNIQUE EXPANSIONS The constant sequences (a 1 ) and (a J ) are always univoque: trivial unique expansions. For decimal expansions: A = {0, 1,..., 9} and q = 10, a sequence c = (c i ) A is univoque if and only if it does not end with 10 or 09. For binary expansions: A = {0, 1} and q = 2, a sequence c = (c i ) A is univoque if and only if it does not end with 10 or 01. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

5 PEDICINI S CHARACTERIZATION Theorem a An expansion (c i ) is unique in a base q 1 + J a 1 max j>1 {a j a j 1 } if and only if the following conditions are satisfied: c n+i a 1 q i < a j+1 a j whenever c n = a j < a J ; a J c n+i q i < a j a j 1 whenever c n = a j > a 1. The theorem implies that a univoque sequence remains univoque in larger bases. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

6 EXPLANATION The conditions ensure that lexicograhically larger sequences lead to larger sums, and lexicograhically smaller sequences lead to smaller sums than (c i ). For instance, the condition c n+i a 1 q i < a j+1 a j whenever c n = a j < a J implies that for any sequence (d i ) first differing from (c i ) at d n > c n we have ( ) ( ) d i c i q i q i a j+1 a j a 1 c n+i q n + q n+i > 0. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

7 TWO-LETTER ALPHABETS Fix A = {0, 1} (without loss of generality) and q > 1. The trivial unique expansions are 0 and 1. Theorem There exist nontrivial unique expansions in base q if and only if q > p := V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

8 PROOF FOR q = p Let (c i ) be a univoque sequence. Then (c i ) cannot contain the blocks 011 and 100 because they are interchangeable: 1 p p 3 = 1 p. Hence if (c i ) is nontrivial, then it must end with (01) But this is impossible either because (01) and 10 are also interchangeable: 1 p p p = 1 p. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

9 PROOF FOR q > p The sequence (c i ) := (10) is univoque because Pedicini s conditions are satisfied: c n+i q i < 1 if c n = 0, 1 c n+i q i < 1 if c n = 1. Indeed, both conditions are equivalent to and thus to q > p. 1 q + 1 q q 5 + < 1 V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

10 A THREE-LETTER ALPHABET Fix A = {0, 1, 2} and q > 1. The trivial unique expansions are 0 and 2. Theorem There exist nontrivial unique expansions in base q if and only if q > 2. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

11 PROOF FOR q = 2 Let (c i ) be a univoque sequence. Then (c i ) cannot contain the blocks 10 and 02 because they are interchangeable: 1 2 = 2 4. For the same reason, it cannot contain the blocks 20 and 12 either. Hence (c i ) = 0 and (c i ) = 2 or (c i ) ends with 1 The latter is also impossible because 1 and 20 are also interchangeable: = 2 2. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

12 PROOF FOR q > 2 The sequence (c i ) := 1 is univoque because Pedicini s conditions are satisfied: c n+i q i < 1 and 2 c n+i q i < 1 for all digits c n = 1. Indeed, both conditions are equivalent to i.e., to q > 2. 1 q + 1 q q 3 + < 1, V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

13 GENERAL THREE-LETTER ALPHABETS Let A = {0, 1, m} with m 2 (without loss of generality) and q > 1. The trivial unique expansions are 0 and m. Theorem For each m 2 there exists a critical base p m > 1 such that there exist nontrivial univoque expansions in all bases q > p m and there are no such expansions in the bases q < p m. Furthermore, the function m p m is continuous and m 2 p m P m := 1 + m 1 for all m. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

14 FURTHER PROPERTIES OF THE CRITICAL BASES Theorem we have p m = 2 if and only if m {2, 4, 8, 16,...}; the set C := {m 2 : p m = P m } is a Cantor set; its smallest element is 1 + x where x is the first Pisot number, i.e., the positive root of the equation x 3 = x + 1. in [2, 1 + x] we have p m = m. all other connected components (m d, M d ) of [2, ) \ C contain a point µ d such that p is strictly decreasing in [m d, µ d ] and strictly increasing in [µ d, M d ]. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

15 DISCOVERY AND PROOF OF THE RESULTS Extensive numerical research of univoque sequences in small bases for fixed integer values of m, based on Pedicini s characterization of univoque sequences. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

16 DISCOVERY AND PROOF OF THE RESULTS Extensive numerical research of univoque sequences in small bases for fixed integer values of m, based on Pedicini s characterization of univoque sequences. These univoque sequences did not contain the smallest digit 0. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

17 DISCOVERY AND PROOF OF THE RESULTS Extensive numerical research of univoque sequences in small bases for fixed integer values of m, based on Pedicini s characterization of univoque sequences. These univoque sequences did not contain the smallest digit 0. It was confirmed theoretically: in bases q P m univoque sequences contain only finitely many zero digits. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

18 DISCOVERY AND PROOF OF THE RESULTS Extensive numerical research of univoque sequences in small bases for fixed integer values of m, based on Pedicini s characterization of univoque sequences. These univoque sequences did not contain the smallest digit 0. It was confirmed theoretically: in bases q P m univoque sequences contain only finitely many zero digits. For most integers m = 2,..., we have found an essentially unique nontrivial univoque sequence of the form (m k 1) with k = [log 2 m] or k = [log 2 m] 1. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

19 DISCOVERY AND PROOF OF THE RESULTS Using Pedicini s characterization we have proved that the sequence (m k 1) is univoque in base q if and only if m ( q 1 1 < π q (m k 1) ) < m 1. The critical value p m was always smaller than P m. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

20 DISCOVERY AND PROOF OF THE RESULTS Using Pedicini s characterization we have proved that the sequence (m k 1) is univoque in base q if and only if m ( q 1 1 < π q (m k 1) ) < m 1. The critical value p m was always smaller than P m. The above inequalities provided the critical base in the whole interval around m in which p m P m. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

21 DISCOVERY AND PROOF OF THE RESULTS Using Pedicini s characterization we have proved that the sequence (m k 1) is univoque in base q if and only if m ( q 1 1 < π q (m k 1) ) < m 1. The critical value p m was always smaller than P m. The above inequalities provided the critical base in the whole interval around m in which p m P m. For some integers m we have found less simple nontrivial univoque sequences, for instance (m 2 1m 2 1m1) for m = 5 and (m 3 1m 2 1) for m = 9. This required the development of a more general theory which allowed us to determine p m for all m 2. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

22 ALGORITHM Given m 2 and P m := 1 + m m 1, there exists a lexicographically largest sequence δ = (δ i ) {1, m} satisfying for all n = 0, 1,..., and 1δ 2 δ 3... δ n+1 δ n+2... δ 1 δ 2 δ 3... δ i P i m m 1. We define p m > 1 by the equation δ i (p m) i = m 1. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

23 ALGORITHM There is a unique truncated sequence δ of δ satisfying δ n+1 δ n+2... δ 1 δ 2... whenever δ n = 1. We define p m > 1 by the equation Then p m = max{p m, p m}. m δ i (p m) i = 1. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

24 EXAMPLE For 2 m < 1 + x we get δ = 1 = δ. Then p m and p m are determined as follows: 1 (p m) i = m 1 = p m = m m 1, m 1 (p m) i = 1 = p m = m. Hence p m = max{p m, p m} = m. V. Komornik (Strasbourg) Generalized golden ratios Liège, June 8, / 19

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