The spectrum of $(\xi {\alpha }^n)$ can be uncountable

In this note, we give a counterexample to the assertion of Problem 10.4 in Bugeaud’s monograph Distribution modulo one and Diophantine approximation, which goes back to Mendès France. The problem states that the spectrum of the sequence $(\xi {\alpha }^n{)}_{n\ge 1}$, that is, the set of irrational $\theta \in (0,1)$ for which $(\xi {\alpha }^n-n\theta {)}_{n\ge 1}$ is not uniformly distributed modulo one, is at most countable for all real $\xi \mathrm{\ne }0$ and $\alpha >1$. More precisely, we prove that for every real $\alpha >1$ there are $2^{{\mathrm{\aleph }}_0}$ real numbers $\xi >0$ for which the spectrum of $(\xi {\alpha }^n{)}_{n\ge 1}$ contains one and the same uncountable set.