On nonnormal numbers
Based on methods of W. M. Schmidt on winning sets, the author gives the following quantitative version of a result of Schmidt (compare also with \textit{S. G. Dani} on orbits of endomorphisms of tori and the Schmidt game [Ergodic Theory Dyn. Syst. 8, 523--529 (1988; Zbl 0673.28009)]). Theorem. There is a positive absolute effective constant \(\kappa> 0\) with the following property: there is an uncountable set of real numbers \(x\) such that \[ \{q^n x\}\leq \exp\Biggl(-\kappa q\Biggl({\ln\overline q\over \ln\ln\overline q}\Biggr)^3\Biggr) \] for any integer \(q\leq 2\) and any nonnegative integer \(n\). (Here \(\{.\}\) stands for the integral part, and \(\overline q= \max\{q, 3\}\).) The right-hand side in the inequality of the theorem does not depend on \(n\).
scientific article; zbMATH DE number 1928270