Density of Some Special Sequences Modulo 1

Abstract

In this paper, we explicitly describe all the elements of the sequence of fractional parts {af(n)/n}, n=1,2,3,…, where f(x)∈Z[x] is a nonconstant polynomial with positive leading coefficient and a≥2 is an integer. We also show that each value w={af(n)/n}, where n≥nf and nf is the least positive integer such that f(n)≥n/2 for every n≥nf, is attained by infinitely many terms of this sequence. These results combined with some earlier estimates on the gaps between two elements of a subgroup of the multiplicative group Zm* of the residue ring Zm imply that this sequence is everywhere dense in [0,1]. In the case when f(x)=x this was first established by Cilleruelo et al. by a different method. More generally, we show that the sequence {af(n)/nd}, n=1,2,3,…, is everywhere dense in [0,1] if f∈Z[x] is a nonconstant polynomial with positive leading coefficient and a≥2, d≥1 are integers such that d has no prime divisors other than those of a. In particular, this implies that for any integers a≥2 and b≥1 the sequence of fractional parts {an/nb}, n=1,2,3,…, is everywhere dense in [0,1].