PreprintCC BY 4.0arXiv
The spectrum of (ξαn) can be uncountable
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 (ξαn)n≥1, that is, the set of irrational θ∈(0,1) for which (ξαn−nθ)n≥1 is not uniformly distributed modulo one, is at most countable for all real ξ≠0 and α>1. More precisely, we prove that for every real α>1 there are 2ℵ0 real numbers ξ>0 for which the spectrum of (ξαn)n≥1 contains one and the same uncountable set.
Related
BibTeX
@misc{ozcan2026spectrum,
title = {{The spectrum of $(ξα^n)$ can be uncountable}},
author = {Hikmet Burak Özcan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07714},
eprint = {2609.07714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07714},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The spectrum of (ξα^n) can be uncountable
AU - Hikmet Burak Özcan
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07714
UR - https://arxiv.org/abs/2609.07714
AB - 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≥1, that is, the set of irrational θ∈(0,1) for which (ξα^n−nθ)_n≥1 is not uniformly distributed modulo one, is at most countable for all real ξ≠0 and α>1. More precisely, we prove that for every real α>1 there are 2^ℵ_0 real numbers ξ>0 for which the spectrum of (ξα^n)_n≥1 contains one and the same uncountable set.
ER -