Uniform distribution of sequences of integers
Ivan Niven · Transactions of the American Mathematical Society · 1961
For example any arithmetic progression { an + b; n =1, 2, 3, } is distributed modulo m if and only if g.c.d. (a, m) = 1. Further we say that the sequence A is distributed in case A is distributed modulo m for every integer m _ 2. The case m = is omitted because A (n, j, 1) = n, and so (1.1) holds for every sequence A in case m = 1. Furthermore the language uniformly distributed modulo 1 has a well-established meaning for a sequence of real numbers I an{o; cf. [1] or [2, p. 72]. These definitions apply to any sequence of integers. Our interest will be primarily in sequences of positive integers satisfying ai<ai+?, at least for i sufficiently large. For such sequences, the definitions can be given in an alternative form. First define G,(n, j, m) as the number of terms ai of the sequence A that satisfy the conditions a _? n and aij(mod m). Also define A (n) as the number of terms ai of A satisfying ai < n, so that A (n) = (i(n, j, 1) for any j. Then the sequence A is distributed modulo m in case