Large sets containing no copies of a given infinite sequence

Kolountzakis, Mihail N.; Papageorgiou, Effie

doi:10.2140/apde.2025.18.93

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Large sets containing no copies of a given infinite sequence

Mihail N. Kolountzakis and Effie Papageorgiou

Vol. 18 (2025), No. 1, 93–108

DOI: 10.2140/apde.2025.18.93

Abstract

Suppose $a_{n}$ is a real, nonnegative sequence that does not increase exponentially. For any $p < 1$ , we construct a Lebesgue measurable set $E \subseteq ℝ$ which has measure at least $p$ in any unit interval and which contains no affine copy ${x + t a_{n} : n \in ℕ}$ of the given sequence (for any $x \in ℝ, t > 0$ ). We generalize this to higher dimensions and also for some “nonlinear” copies of the sequence. Our method is probabilistic.

Keywords

Erdős similarity problem, Euclidean Ramsey theory, probabilistic method

Mathematical Subject Classification

Primary: 05D40, 28A80

Milestones

Received: 4 August 2022

Revised: 19 August 2023

Accepted: 3 October 2023

Published: 15 December 2024

Authors

Mihail N. Kolountzakis
Department of Mathematics and Applied Mathematics University of Crete Heraklion Greece

Effie Papageorgiou
Department of Mathematics and Applied Mathematics University of Crete Heraklion Greece

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2025, the article will become Open Access in early 2025. Otherwise, this article (and all 2025 articles) will be available only to paid subscribers.

Vol. 18, No. 1, 2025