Distribution modulo 1 of some oscillating sequences
The distribution mod 1 of sequences of the form P(n)F(Q(n)), \(n=1,2,..\). is studied, where P and Q are polynomials and F is a sufficiently many times differentiable periodic function. For example Theorem 2.2 states that for any given nonnegative integers d and e, there exist numbers \(s=s(d,e)\) and \(r=r(d,e)>0\) possessing the following property: For any polynomials P and Q of degrees d and e, respectively, and any periodic function F which is s times differentiable at Q(0) the inequality \[ \| P(n)F(Q(n))-P(0)F(Q(0))\| <n^{-r} \] has infinitely many positive integer solutions. When Q is restricted being a monomial then a minor extra condition on F provides that the values P(n)F(Q(n)) are dense mod 1 while for linear Q and smooth F those values are uniformly distributed mod 1.
scientific article; zbMATH DE number 4198101
scientific article