About the fractional parts of the powers of the rational numbers

Details About the fractional parts of the powers of the rational numbers About the fractional parts of the powers of the rational numbers

2006-11-20

arxiv.org/abs/math/0611622v1

Mathematics

Number Theory

5 pages

Scientific paper

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.

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