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 $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for which the set of positive real numbers $\lambda$ satisfying $<\lambda (p / q)^n> \in A(\epsilon)$ $(\forall n \in \mathbb{N})$ is uncountable.