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Almost Erdős sets

Part of: Classical measure theory Connections with other structures, applications Dynamical systems and ergodic theory

Published online by Cambridge University Press:  12 February 2026

Arian Bërdëllima Open the ORCID record for Arian Bërdëllima [Opens in a new window]
Arian Bërdëllima*
Affiliation:
Faculty of Engineering, German International University in Berlin, Am Borsigturm 162, Berlin 13507, Germany (berdellima@gmail.com)
*
*Corresponding author.
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Abstract

A set $S \subseteq \mathbb{R}$ is almost Erdős if, for every $\varepsilon \gt 0$, there exists a set $E \subseteq \mathbb{R}$ of positive Lebesgue measure such that $\{x \in S : ax+b \notin E\}$ is nonempty for all $|a| \gt \varepsilon$ and $b \in \mathbb{R}$. In this note, we show that any decreasing null sequence $(x_n)$ with decay rate greater than $1/2$ is an almost Erdős set.

Keywords

almost Erdős setsdecreasing null sequencesLebesgue measureHausdorff dimensionadditive combinatorics

MSC classification

Primary: 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Secondary: 28A80: Fractals 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 37-00: General reference works (handbooks, dictionaries, bibliographies, etc.)

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 9
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

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