On an Erdős similarity problem in the large
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909643538694144 |
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| author | Gao, Xiang Mooroogen, Yuveshen Yip, Chi Hoi |
| author_facet | Gao, Xiang Mooroogen, Yuveshen Yip, Chi Hoi |
| contents | In a recent paper, Kolountzakis and Papageorgiou ask if for every $ε\in (0,1]$, there exists a set $S \subseteq \mathbb{R}$ such that $\vert S \cap I\vert \geq 1 - ε$ for every interval $I \subset \mathbb{R}$ with unit length, but that does not contain any affine copy of a given increasing sequence of exponential growth or faster. This question is an analogue of the well-known Erdős similarity problem. In this paper, we show that for each sequence of real numbers whose integer parts form a set of positive upper Banach density, one can explicitly construct such a set $S$ that contains no affine copy of that sequence. Since there exist sequences of arbitrarily rapid growth that satisfy this condition, our result answers Kolountzakis and Papageorgiou's question in the affirmative. A key ingredient of our proof is a generalization of results by Amice, Kahane, and Haight from metric number theory. In addition, we construct a set $S$ with the required property -- but with $ε\in (1/2, 1]$ -- that contains no affine copy of $\{2^n\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06727 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On an Erdős similarity problem in the large Gao, Xiang Mooroogen, Yuveshen Yip, Chi Hoi Classical Analysis and ODEs Number Theory 28A75, 28A78 (Primary) 28A80, 11K55, 11J71, 11B05 (Secondary) In a recent paper, Kolountzakis and Papageorgiou ask if for every $ε\in (0,1]$, there exists a set $S \subseteq \mathbb{R}$ such that $\vert S \cap I\vert \geq 1 - ε$ for every interval $I \subset \mathbb{R}$ with unit length, but that does not contain any affine copy of a given increasing sequence of exponential growth or faster. This question is an analogue of the well-known Erdős similarity problem. In this paper, we show that for each sequence of real numbers whose integer parts form a set of positive upper Banach density, one can explicitly construct such a set $S$ that contains no affine copy of that sequence. Since there exist sequences of arbitrarily rapid growth that satisfy this condition, our result answers Kolountzakis and Papageorgiou's question in the affirmative. A key ingredient of our proof is a generalization of results by Amice, Kahane, and Haight from metric number theory. In addition, we construct a set $S$ with the required property -- but with $ε\in (1/2, 1]$ -- that contains no affine copy of $\{2^n\}$. |
| title | On an Erdős similarity problem in the large |
| topic | Classical Analysis and ODEs Number Theory 28A75, 28A78 (Primary) 28A80, 11K55, 11J71, 11B05 (Secondary) |
| url | https://arxiv.org/abs/2311.06727 |