Non-Salem Sets in Metric Diophantine Approximation
Publication Date
7-1-2023
Document Type
Article
Publication Title
International Mathematics Research Notices
Volume
2023
Issue
15
DOI
10.1093/imrn/rnac206
First Page
13136
Last Page
13152
Abstract
A classical result of Kaufman states that, for each τ > 1, the set of τ-well approximable numbers {equation presented} is a Salem set. A natural question to ask is whether the same is true for the sets of τ -well approximable n × d matrices when nd > 1 and t > d/n. We prove the answer is no by computing the Fourier dimension of these sets. In addition, we show that the set of badly approximable n × d matrices is not Salem when nd > 1. The case of nd = 1, that is, the badly approximable numbers, remains unresolved.
Funding Number
803711
Funding Sponsor
Horizon 2020 Framework Programme
Department
Mathematics and Statistics
Recommended Citation
Kyle Hambrook and Han Yu. "Non-Salem Sets in Metric Diophantine Approximation" International Mathematics Research Notices (2023): 13136-13152. https://doi.org/10.1093/imrn/rnac206
