Diophantine approximations with arithmetic restrictions and an application to trigonometric functions (Q2774477)

Let \(\xi\) be irrational and \(a\), \(s\) be integers with \(0<a<s\). It is proved that there are infinitely many fractions \(u/v\) with \(u\equiv a\bmod s\) satisfying \(|\xi-u/v|<s/(4v^2)\). Lower bounds for \(|\xi-u/v|\) can be proved for certain irrationals with specific continued fraction expansions. Applying these results, lower bounds for the greatest accumulation points of the sequences \(|\sin(\pi\alpha n)|^{n^2}\) and \(|\cos(\pi\alpha n)|^{n^2}\) are computed for irrationals \(\alpha\). These sequences of trigonometric functions provide a way to decide whether the continued fraction expansion of an irrational \(\alpha\) is bounded or not.