On the Eigen-Falconer theorem in $\mathbb{R}^d$
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918227470188544 |
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| author | Li, Wenxia Wang, Zhiqiang Xu, Jiayi |
| author_facet | Li, Wenxia Wang, Zhiqiang Xu, Jiayi |
| contents | In this paper, we study the analogous Erdős similarity conjecture in higher dimensions and generalize the Eigen-Falconer theorem. We show that if $A=\{\boldsymbol{x}_n\}_{n=1}^\infty \subseteq \mathbb{R}^d$ is a sequence of non-zero vectors satisfying \[ \lim_{n \to \infty} \|\boldsymbol{x}_n\| =0 \quad \text{and} \quad \lim_{n \to \infty} \frac{\|\boldsymbol{x}_{n+1}\|}{\|\boldsymbol{x}_n\|} = 1, \] then there exists a measurable set $E \subseteq \mathbb{R}^d$ with positive Lebesgue measure such that $E$ contains no affine copies of $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Eigen-Falconer theorem in $\mathbb{R}^d$ Li, Wenxia Wang, Zhiqiang Xu, Jiayi Classical Analysis and ODEs 28A75 In this paper, we study the analogous Erdős similarity conjecture in higher dimensions and generalize the Eigen-Falconer theorem. We show that if $A=\{\boldsymbol{x}_n\}_{n=1}^\infty \subseteq \mathbb{R}^d$ is a sequence of non-zero vectors satisfying \[ \lim_{n \to \infty} \|\boldsymbol{x}_n\| =0 \quad \text{and} \quad \lim_{n \to \infty} \frac{\|\boldsymbol{x}_{n+1}\|}{\|\boldsymbol{x}_n\|} = 1, \] then there exists a measurable set $E \subseteq \mathbb{R}^d$ with positive Lebesgue measure such that $E$ contains no affine copies of $A$. |
| title | On the Eigen-Falconer theorem in $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs 28A75 |
| url | https://arxiv.org/abs/2512.02146 |