Density of Some Special Sequences Modulo 1

In this paper, we explicitly describe all the elements of the sequence of fractional parts {a(f (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 = {a(f (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 Z*m 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 {a(f (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/ bvn}, n = 1, 2, 3, ... , is everywhere dense in [0, 1].