On the fractional parts of a n / n

We give various results about the distribution of the sequence { a n / n } n ⩾ 1 modulo 1, where a ⩾ 2 is a fixed integer. In particular, we find an explicit infinite subsequence such that { a n / n } n ∈ is uniformly distributed modulo 1. Also we show that for any constant c > 0 and a sufficiently large N , the fractional parts of the first N terms of this sequence hit every interval ⊆ [0, 1] of length || ⩾ c N −0.475 .