Dependence in Probability, Analysis and Number Theory Berkes, Bradley, Dehling, Peligrad, Tichy (editors) c  Kendrick Press 2010, pp. 51–72 Weyl’s theorem in the measure theory of numbers Roger C. Baker, Ryan Coatney and Glyn Harman Dedicated to the memory of Walter Philipp Abstract Let S = {a 1 , a 2 ,...} be a real sequence, a k+1 - a k ≥ σ > 0 (k = 1, 2,...). Weyl proved in 1916 that the sequence S x : a 1 x, a 2 x, a 3 x,... is uniformly distributed (mod 1) for almost all x ∈ R. Interpreted in the obvious way, this remains valid if x varies over R d . The following results are proved about the null set E (d) (S )= {x ∈ R d : S x is not uniformly distributed (mod 1)}. (i) If a k = O(k p ), then E (d) (S ) has dimension ≤ d - 1/ p. Given p, this bound is attained for suitable S . (ii) The intersection of E (d) (S ) with a curve C satisfying natural conditions is a null subset of C , and has dimension ≤ 1 - 1/ pd if a k = O(k p ). (iii) Let d = 1. Suppose a k ∈ N (k ≥ 1), a k ≤ Ck for infinitely many k. The subset B δ (S ) of E (d) (S ) ∩ [0, 1) consisting of x for which S x has bias b(x) ≥ δ > 0 (defined below) is finite. A cardinality bound is given, and is strengthened for the set H I (S )= {x : S x omits the interval I (mod 1)}. Roger C. Baker Department of Mathematics, Brigham Young University, Provo, UT 84602, USA, e-mail: baker@math.byu.edu Ryan Coatney Department of Mathematics, Brigham Young University, Provo, UT 84602, USA, e-mail: r.coatney@gmail.com Glyn Harman Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, England, e-mail: g.harman@rhul.ac.uk 51