Notes on Uniform Distribution Modulo One

We exhibit a sequence (u_n) which is not uniformly distributed modulo one even though for each fixed integer k > 2 the sequence (ku_n) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit a sequence (u_n) which is u.d. (mod 1) even though no subsequence of the form {u_kn+j) is u.d. (mod 1) for any k > 2. We prove that, if the subsequences (u_kn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set p, then (u_n) is u.d. (mod 1) if the sum of the reciprocals of the primes in p diverges. We show that this result is the best possible of its type.