Density of Some Special Sequences Modulo 1

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