The diaphony of a class of infinite sequences

Abstract

Let $\alpha$ be an irrational number with diophantine approximation properties and let $\ell (z)$ be a logarithmic-like function. We study the diaphony $F_N$ of the sequence $(\alpha n + \ell (n))_{n \ge 1}$. As an example of our result, we show that if $\alpha$ has the bounded partial quotients of the continued fraction expansion and $\beta$ is non-zero real, then the sequence $(x_n)_{n \ge 1} = (\alpha [(n+1)/2] + (-1)^{n+1} \beta \log ([(n+1)/2]))_{n \ge 1}$ satisfies $N^{-\frac{3}{4}-\varepsilon} \ll F_N (x_n) \ll N^{-\frac{2}{3}}$ for any $0 \lt \varepsilon \lt 1/4$. In our proof, Atkinson's saddle-point lemma is very useful.