About the fractional parts of the powers of the rational numbers

Abstract

Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $ε> 0$, there exists a set $A(ε) \subset [0, 1[$, with finite border and with Lebesgue measure $< ε$, for which the set of positive real numbers $λ$ satisfying $<λ(p / q)^n> \in A(ε)$ $(\forall n \in \mathbb{N})$ is uncountable.