Distribution modulo 1 of some oscillating sequences. II (Q1905776)

The distribution modulo one of sequences defined by polynomials and quasi--periodic functions were studied in \textit{D. Berend} and \textit{G. Kolesnik} [Isr. J. Math. 71, 161-179 (1990; Zbl 0726.11042)]. In this paper some of those results are improved and generalized. The general theorems are too long and complicated to state, so we illustrate them here by some examples. For every positive integer \(r\) there exists a \(\rho=\rho(r)\) such that for any real numbers \(\alpha_1,\dots,\alpha_r\) the inequality \[ |n\cos(n\alpha_1)\dots\cos(n\alpha_r)|<n^{-\rho} \] has infinitely many solutions in the integer \(n\). Suppose \(f\) is periodic, non--constant, twice differentiable at 0 with \(f'(0)=0\) and \(\alpha\) is any real number. There is a constant \(C>0\) such that the inequality \[ |nf(n\alpha)|<Cn^{-1/3} \] has infinitely many solutions in the integer \(n\). Let \(P\) be a real polynomial of degree \(d\) and \(\alpha\) be a real number. If \(f\) is a 1--periodic \(C^d\) smooth function then the inequality \[ |P(n)f(n\alpha)-P(0)f(0)|<\epsilon \] has infinitely many solutions for every \(\epsilon>0\). On the other hand, if \(\alpha\) is not too well approximable then there exists a 1--periodic \(C^{d-1}\) smooth function \(f\) such that \[ |P(n)f(n\alpha)-P(0)f(0)|=\frac 12 \] for \(n=1,2,\dots\).