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 ≥ n f and n f is the least positive integer such that f ( n ) ≥ n / 2 for every n ≥ n f , 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 Z m 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 ) / n d } , 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 { a n / n b } , n = 1 , 2 , 3 , … , is everywhere dense in [ 0 , 1 ] .