Year: 2026
Type: preprint
Abstract: 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 $(ξα^n)_{n\ge1}$, that is, the set of irrational $θ\in(0,1)$ for which $(ξα^n-nθ)_{n\ge1}$ is not uniformly distributed modulo one, is at most countable for all real $ξ\ne0$ and $α>1$. More precisely, we prove that for every real $α>1$ there are $2^{\aleph_0}$ real numbers $ξ>0$ for which the spectrum of $(ξα^n)_{n\ge1}$ contains one and the same uncountable set.
Source: arXiv (Cornell University)
Author: Özcan, Hikmet Burak
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212005728