Small values of \(n^ 2\alpha\pmod 1\)
We prove that for any real number \(\alpha\), any \(\varepsilon >0\) and any \(N\geq 1\) one has: \[ \min_{1\leq n\leq N} |n^2 \alpha|\;\ll_\varepsilon\;N^{-(4/7)+ \varepsilon}. \] Moreover, \(|n^2 \alpha|< n^{-(2/3)+ \varepsilon}\) for infinitely many positive integers \(n\). Here \(|\cdot |\) stands for the distance to the nearest integer and the implied constant in the first inequality depends only on \(\varepsilon\).
scientific article; zbMATH DE number 807309
scientific article