ILL 1987 Problem 64

Let $r > 1$ be a real number, and let $n$ be the largest integer smaller than $r$ . Consider an arbitrary real number $x$ with $0 \leq x \leq \frac{n}{r-1}.$ By a base- $r$ expansion of $x$ we mean a representation of $x$ in the form $$x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots$$ where the $a_i$ are integers with $0 \leq a_i < r.$ You may assume without proof that every number $x$ with $0 \leq x \leq \frac{n}{r-1}$ has at least one base- $r$ expansion.

Prove that if $r$ is not an integer, then there exists a number $p$ , $0 \leq p \leq \frac{n}{r-1}$ , which has infinitely many distinct base- $r$ expansions.

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