Dr. Bergelson s Bonus Problems

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1 Dr. Bergelson s Bonus Problems Honors Undergraduate Number Theory The Ohio State University, Winter & Spring 2012 compiled by: Daniel Glasscock Caution: some of these problems may be extremely difficult! 1. Find infinitely many integer solutions to the equation x 2 2y 2 = Prove that the square is the only regular polygon inscribable in Z Prove that regular 3-, 4-, and 6-gons are the only regular polygons inscribable in Z d, d Show that the regular tetrahedron is embeddable in Z Which regular tetrahedron are embeddable in Z d, d 3? 6. Show that e 2 is irrational using the fact that e = n 1/n!. 7. Show that the following sets are not AP-rich: {p(n) n N} where p Z[x], deg(p) 2 {z n z Z, n N} 8. Show that AP-rich sets are partition regular. 9. Prove that if d(a) > 0, then a A a 1 = ; show that the converse is not true. 10. Give an example of an AP-rich set A N for which a A a 1 < and d(a) = Show that the following sets of positive integers have zero density: squares, sums of two squares, primes, Fibonacci numbers, FS(3 n ) n= Show that if d(a) = d and A = r i A i, then some A i satisfies d(a i ) d/r. 13. Show that there is an A N such that d(a) = 0, d(a) = Show that there is a partition N = A B such that d(a) = d(b) = Show that there is a partition N = i A i such that d(a i ) = 1 for all i. 16. Show that there are A, B N such that d(a), d(b) exist while d(a B), d(a B) do not. 17. Show that if C i = a i N + b i are disjoint in N, then d( r i C i) = r i a 1 i. 18. Show that if A = N \ {n is written with a 0}, then a A a 1 <. 19. Prove that if A R, m(a) > 1, then (A A) N. 20. Give a coloring of N in which x + y = 3z has no monochromatic solution. 21. Prove that the system of equations x 2 x 1 = x 0,..., x n x n 1 = x 0 is partition regular. 22. Prove that quotients of primes are dense in the positive real numbers. 23. Prove that Q + 2Q and Q + 3Q are isomorphic as vectorspaces over Q but not as fields. 24. Show that Q( {( ) 2) a 2b = a, b Q} and Z/5Z( {( ) 2) b a a 2b = a, b Z/5Z}. b a 25. Prove that if 2 is not a quadratic residue modulo a prime p, then {a + 2b a, b Z/pZ} is a field of p 2 elements.

2 26. Find infinite order, non-isomorphic fields of characteristic p. 27. Construct a field of 8 elements. 28. Prove that for all n N, there exists a circle C n R 2 which intersects Z 2 in at least n points. 29. How many infinite binary sequences are there without two consecutive 0 s? 30. Prove that n F n/2 n+1 = 1 where F n is the n-th Fibonacci number. 31. Formulate a finitistic version of van der Waerden s theorem, and show that it is equivalent to the infinitistic formulation. 32. Give an example of a finite coloring of N such that there is no monochromatic configuration {n, 2n}. 33. Show that if the primes are partitioned r i C i, then some C i contains a solution to the equation x + y + z = 3w. 34. Formulate a finitistic version of Szemerédi s theorem, and show that it is equivalent to the infinitistic formulation. 35. Prove that if d(a) > 1/2, then A A = N. 36. Show that d(square-free) = 6/π 2 ; conclude that every number is the difference of two square-free numbers. 37. Show that every integer is the sum of two square-free numbers. 38. Is it true that A + A A A for all finite A Z? 39. Is it true that if d(a) > 1/2, then N \ (A + A) <? 40. Find infinitely many quadratic polynomials p(x) Z[x] such that the following generalization of Sárközy s theorem fails to hold: If d(a) > 0, then (A A) {p(n) n N}. 41. Formulate a finitistic version of Sárközy s theorem, and show that it is equivalent to the infinitistic formulation. 42. Show that the following strengthening of Sárközy s theorem is equivalent to the original: If d(a) > 0, then (A A) {n 2 n N} =. 43. Prove that if d(a) > 0, then A A is syndetic. 44. Prove that if A N, B Z, B = k, and A + B = N, then d(a) 1/k. 45. Prove that if A N is syndetic, then d(a) > Show that if A N is syndetic, then A \ {n 2 n N} is syndetic. 47. Prove that if d(a) > 0, then there is an arbitrarily large n N such that d(a (A n)) > Prove that if A Z n, d(a) > 1/2, then A A = Z n. 49. Prove that if A Z n, d(a) > 0, then A A is syndetic. 50. Is A Z 2 necessarily syndetic if A ({n} Z) and A (Z {n}) are syndetic for every n Z? 51. Give an example of A 1, A 2 Z with d(a 1 ), d(a 2 ) > 0 such that d(a 1 A 2 ) = 0; redefine d for subsets of Z 2 to correct for this. 52. Show that if E n R has measure zero for each n N, then n E n has measure zero. 53. Show that Cantor s middle thirds set is an uncountable set with measure zero.

3 54. If C is Cantor s middle thirds set, show that C + C = [0, 2], C C = [ 1, 1]. 55. Show that Cantor s middle thirds set contains a Hamel base. 56. Find a Cantor set whose set of difference does not contain any interval. 57. Show that if α / Q, then (n 2 α) n is dense modulo Find and prove necessary and sufficient conditions on α, β for (n 2 α + nβ) n to be dense modulo Show that for 0 < c <, c / N, (n c ) n is dense modulo Give an example of a sequence (x n ) n which is not dense modulo 1 but for which (x n+k x n ) n is dense for each k N. 61. Use the fact that ( n) n is uniformily distributed modulo 1 to show that (n n) n is as well. 62. Enumerate the rational numbers (r n ) n = Q such that (r n ) n is uniformily distributed modulo Prove that x R is normal in base 2 if and only if (2 n x) n is uniformily distributed modulo Find a sequence (x n ) n which is dense but not uniformily distributed modulo Show that if x n y n 0 and x n is uniformily distributed, then y n is also uniformily distributed. 66. Show that every real number is the difference of two normal numbers. 67. Show that the following statement is true for each of the indicated function spaces L: the sequence 1 N (x n ) n [0, 1] is uniformily distributed if and only if for all f L, N n f(x n) 1 fdx as 0 N. L = step functions on [0,1] L = C[0, 1] L = Riemann integrable functions on [0,1] L = R[x] L = trigonometric polynomials 68. Let M = {0, 1} N and d 1, d 2 : M M R be defined by { 0 if x = y d 1 (x, y) =, d 2 (x, y) = 1/i if i is minimal so x i y i Show that d 1, d 2 are distinct metrics on M which generate the same topology. 69. Prove that the transformation T (x) = 2x of R/Z is ergodic. i=1 x i y i 2 i. 70. Prove that the transformation {{ 1 } T (x) = x if x (0, 1] 0 if x = 0 of R/Z preserves the measure induced by l([a, b]) = (log 2) 1 b a (1 + x) 1 dx. 71. Show that the elements of { p p prime} are linearly independent over Q. 72. Show that if f : R R satisfies f(x + y) = f(x) + f(y) for all x, y R, then either f(x) = cx for some c R or the graph of f is dense in R Formulate Szemerédi s theorem, Sárközy s theorem, and the theorem on the syndeticity of A A using upper Banach density, and show their equivalence to the original theorems.

4 74. For α / Q, k N, show that (n k α) n is dense modulo 1 using the k-dimensional formulation of van der Wareden s theorem. 75. Show that there exists an α / Q such that {n 2 α m 2 n, m Z} is not dense. 76. Find a Cantor set in R which is algebraically independent over Q. 77. Find a Cantor set C R such that (C C) Q =. 78. Prove this recursive formula for ω n, the volume of the unit ball in R n : ω 1 = 2, ω 2 = π, ω n = 2π n ω n Show that primes p of the form 3k + 1 can be represented by the form x 2 + 3y Show that primes p of the form 8k + 1 can be represented by the form x 2 + 2y Describe the integers represented by the forms 2x 2 + 3y 2 and x 2 + 5y Improve Dirichlet s pigeon-hole argument to achieve the following: for all irrational α R, there exist infinitely many p q Q such that α p q < 1 2q Let α R. Show that there are only finitely many p q Q such that α p q < 1 5q 2 if and only if the digits in the continued fraction expansion of α are eventually all Prove that {n 2 + (n + 1) 2 n N} N 2 =. 85. Find an example of a geometric problem leading to Pell s equation. 86. Given two rational points on an ellipse, are there infinitely many? 87. Characterize the circles C in R 2 for which C Q 2 =. 88. Let M be a unimodular matrix (that is, M is an integer matrix with det M = ±1). Prove that M 1 is unimodular. 89. Show that the equation x + y = z 2 is partition regular. 90. Call a set S N Sárközy good if for all A N with d(a) > 0, there exists an n S such that d(a (A n 2 )) > 0. Show that 17N is Sárközy good. Show that if C i is a finite partition of N, then some C i is Sárközy good. What other subsets of N are Sárközy good? 91. Prove that π 2 /6 = (1 xy) 1 dxdy. 92. Prove that ζ(n) = [0,1] n (1 x 1 x n ) 1 dx 1 dx n. 93. Endow {1, 2,..., n} N with a natural metric and describe the induced topology. 94. For all n, m N, prove that {1, 2,..., n} N is homeomorphic to {1, 2,..., m} N. 95. Endow {1, 2,..., n} Z with a natural metric, and prove that {1, 2} N is homeomorphic to {1, 2} Z. 96. Endow N N with a natural metric, and prove that {1, 2} N is homeomorphic to N N. 97. Show that {1, 2} N is homeomorphic to Cantor s middle thirds set. 98. Generalize Cantor s construction, and show that {1, 2} N is homeomorphic to generalized Cantor sets. 99. Show that a Cantor set is the disjoint union of uncountably many Cantor sets.

5 100. Let A N be normal (associated to a normal binary number). Show that for all integers n 1 < < n k, d(a (A n 1 ) (A n k )) = 2 (k+1) Fix A N, and define ϕ 2 (n) = d(a (A n)). Show that (ϕ 2 (n)) n is positive definite (that is, for all {ξ i } N i= N C, N m,n= N ϕ 2(n m)ξ n ξ m 0) Describe the set of maximal ideals of the following rings: C[0, 1], C(R), C[x], C[x 1,..., x n ] Let B 0 be the set of continuous, real valued functions on [0, 1], and for each n N, let B n be the set of functions which are pointwise limits of functions in B n 1. Show that χ Q B 2 \ B Show that the only norms on fields of positive characteristic are non-archimedean Prove that the following three rings are all isomorphic (this ring is Hamilton s quaternions): {a + bi + cj + dk a, b, c, d R, ij = k, jk = i, ki = j, i 2 = j 2 = k 2 = 1} { ( ) ( ) ( ) ( ) 1 0 i i a + b + c + d a, b, c, d R} i 1 0 i 0 {( ) z w w, z C} w z 106. Arrange the multiplication table of a finite group G so that the identity element is the only element along the diagonal. Let ρ(g) be the 0-1 matrix corresponding to the location of g in the multiplication table. Show that ρ is a representation of G. Does this work for infinite groups? 107. Prove that the identity map is the only automorphism of R Prove that the identity map is the only automorphism of Q p Find infinitely many automorphisms of C Prove that the Q p s and R are non-isomorphic fields Prove that Q p is homeomorphic to Q q Characterize rational numbers in terms of their decimal expansions, p-adic expansions, and continued fraction expansions Use the ergodic theorem and the continued fraction map to show that almost all real numbers (with respect to both the Lebesgue and Gauss measures) have all positive integers in their continued fraction expansions. Find their frequencies Let (p n ) n be the sequence of primes. Give an elementary proof that for all irrational α, (p n α) n is dense modulo Find uncountably many irrational α such that (x n ) n is not dense modulo Prove that a smooth function on R is polynomial if and only if it is zero after finitely many iterations of d/dx Prove that a function f on Z is polynomial if and only if it is zero after finitely many iterations of (f)(n) = f(n + 1) f(n) Prove that a function f on Z is a quadratic polynomial if and only if for all n, m, k Z, f(n + m + k) f(n + m) f(n + k) f(m + k) + f(n) + f(m) + f(k) f(0) = 0. Generalize this. Make a similar statement using values of f along arithmetic progressions Prove that for all x Q, p x p = It is possible to split a square with side length 1 into 4 disjoint pieces of equal area with (not necessarily straight) lines of total length strictly less than 2?