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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.01279 |
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| _version_ | 1866911237657329664 |
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| author | Kiss, Gergely Laczkovich, Miklós |
| author_facet | Kiss, Gergely Laczkovich, Miklós |
| contents | Let $K$ be a nonempty finite subset of the Euclidean space $\mathbb{R}^k$ $(k\ge 2)$. We prove that if a function $f\colon \mathbb{R}^k\to \mathbb{C}$ is such that the sum of $f$ on every congruent copy of $K$ is zero, then $f$ vanishes everywhere. In fact, a stronger, weighted version is proved. As a corollary we find that every finite subset $K$ of $\mathbb{R}^k$ having at least two elements is a Jackson set; that is, no subset of $\mathbb{R}^k$ intersects every congruent copy of $K$ in exactly one point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions to the discrete Pompeiu problem and to the finite Steinhaus tiling problem Kiss, Gergely Laczkovich, Miklós Functional Analysis Metric Geometry Spectral Theory 30D05, 43A45, 52C99 Let $K$ be a nonempty finite subset of the Euclidean space $\mathbb{R}^k$ $(k\ge 2)$. We prove that if a function $f\colon \mathbb{R}^k\to \mathbb{C}$ is such that the sum of $f$ on every congruent copy of $K$ is zero, then $f$ vanishes everywhere. In fact, a stronger, weighted version is proved. As a corollary we find that every finite subset $K$ of $\mathbb{R}^k$ having at least two elements is a Jackson set; that is, no subset of $\mathbb{R}^k$ intersects every congruent copy of $K$ in exactly one point. |
| title | Solutions to the discrete Pompeiu problem and to the finite Steinhaus tiling problem |
| topic | Functional Analysis Metric Geometry Spectral Theory 30D05, 43A45, 52C99 |
| url | https://arxiv.org/abs/2403.01279 |