On uniform distribution and the density of sum sets

Abstract

1. For any lattice point a = (a1, , ak) in the k-dimensional euclidean space Rk, let |a|=max,,-,...,k Ia,. If A is an infinite set of such lattice points, we define for x>0 the counting function A (x) to be the number of elements aEA satisfying Ilall x. Then various densities for such sets A can be introduced as generalizations of the well-known densities of sets of non-negative integers. In particular, we shall denote the lower limit, the upper limit, and, in the case of its existence, the limit, of the sequence A(x)/(2x)k, as x tends to infinity, by d*(A), d*(A), and d(A), respectively. Furthermore, if A is restricted to have elements with non-negative coordinates only, we shall consider the corresponding expressions of the sequence A (x)/xk and denote them by D*(A), D*(A), and D(A). According to the terminology in the case k =1, we shall call these limits the lower and upper asymptotic densities and the natural density1 of A, respectively. The sum set A +B of two sets A and B in Rk is, as usual, defined to be the set of all points a+b, aEA, bEB, obtained by vector addition. By an interval ICRk we mean the cartesian product of any k open intervals of RI. The unit cube, i.e. the set of all points g = (xi, * * *, Xk) with O<x,,<1 (K = 1, * * *, k), is denoted by C*. For any real number x, let [x] denote the greatest integer < x and let {x} be the fractional part x-[x]. Then, for every point = (xl, . . *, 9Xk), let {I)} ({x} ,*..., Xk} ), and for every set MCRk, let { M} be the set of all points { with TEM. The Jordan content of a set MCRk is denoted by Ak(M). A sequence X, 2, . . . is called uniformly distributed (mod 1) if it has the following property: Let, for any interval ICCk, N1 be the set of indices i such that {,} CI. Then N1 has a natural density D (NI) =AA;(1)