A structure formula for the sequence \(\{n\theta\}\) and some of its applications in problems of number theory (Q2862892)

Let \(\{\alpha\}\) be the fractional part of a real number \(\alpha\). In this paper, the author studies the fractional parts \(\{n \theta\}_{n=1}^{\infty}\) for some quadratic numbers \(\theta\) and some \(\theta\) expressible by some special continued fractions. In particular, he proves that the remainder term \(O(\log x)\) in the asymptotic formula NEWLINE\[NEWLINE\sum_{n \leq x} \{n \theta\}-x/2=O(\log x)NEWLINE\]NEWLINE for so-called Chebyshev numbers \(\theta\) is best possible.