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ON THE DISTRIBUTION OF SEQUENCES OF THE FORM (qny)

Kristensen, Simon and Persson, Tomas LU orcid (2025) In Mathematica Scandinavica 131(1). p.17-34
Abstract

We study the distribution of sequences of the form (qny)∞n=1, where (qn)∞n=1 is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points γ ∈ [0, 1) which are well approximated by points in the sequence (qny)∞n=1. The bounds on Hausdorff dimension are valid for almost every y in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is... (More)

We study the distribution of sequences of the form (qny)∞n=1, where (qn)∞n=1 is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points γ ∈ [0, 1) which are well approximated by points in the sequence (qny)∞n=1. The bounds on Hausdorff dimension are valid for almost every y in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is itself Lebesgue measure, our measure bounds are also sharp for a very large class of sequences.

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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematica Scandinavica
volume
131
issue
1
pages
18 pages
publisher
Mathematica Scandinavica
external identifiers
  • scopus:105001290130
ISSN
0025-5521
DOI
10.7146/math.scand.a-151576
language
English
LU publication?
yes
id
4a4d7995-9536-4316-ac94-545f63be4af4
date added to LUP
2025-09-11 10:18:45
date last changed
2025-09-25 15:52:11
@article{4a4d7995-9536-4316-ac94-545f63be4af4,
  abstract     = {{<p>We study the distribution of sequences of the form (q<sub>n</sub>y)<sup>∞</sup><sub>n</sub><sub>=1</sub>, where (q<sub>n</sub>)<sup>∞</sup><sub>n</sub><sub>=1</sub> is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points γ ∈ [0, 1) which are well approximated by points in the sequence (q<sub>n</sub>y)<sup>∞</sup><sub>n</sub><sub>=1</sub>. The bounds on Hausdorff dimension are valid for almost every y in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is itself Lebesgue measure, our measure bounds are also sharp for a very large class of sequences.</p>}},
  author       = {{Kristensen, Simon and Persson, Tomas}},
  issn         = {{0025-5521}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{17--34}},
  publisher    = {{Mathematica Scandinavica}},
  series       = {{Mathematica Scandinavica}},
  title        = {{ON THE DISTRIBUTION OF SEQUENCES OF THE FORM (q<sub>n</sub>y)}},
  url          = {{http://dx.doi.org/10.7146/math.scand.a-151576}},
  doi          = {{10.7146/math.scand.a-151576}},
  volume       = {{131}},
  year         = {{2025}},
}