On fractional parts of powers of real numbers close to 1

Abstract

We prove that there exist arbitrarily small positive real numbers $\epsilon$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} ε|\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17}|\log \epsilon {|}^{-1}$. We also establish that the set of real numbers $\alpha >1$ such that the sequence of fractional parts $(\{{\alpha }^n\}{)}_{n\ge 1}$ is not dense modulo 1 has full Hausdorff dimension.

12 pages

References in corpus (2)

• A version of the proof for Peres-Schlag's theorem on lacunary sequences
• Density modulo 1 of sublacunary sequences: application of Peres-Schlag's arguments