About the fractional parts of the powers of the rational numbers

Abstract

Let $p/q$ ($p,q\in {\mathbb{N}}^{\ast }$) 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.

5 pages