Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866908906462117888 |
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| author | Polasek, Iván Rela, Ezequiel |
| author_facet | Polasek, Iván Rela, Ezequiel |
| contents | In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This builds upon Mockenhaupt's Restriction Theorem; while this theorem could only be applied to sets of positive Fourier dimension, we show that the existence of a measure with polylogarithmic Fourier decay combined with full Hausdorff dimension 1 on the real line is enough to guarantee restriction. In order to achieve this, we combine two different tools: a modification of a construction from a recent work of Li and Liu to produce a set with specific Hausdorff and Fourier dimensions, and a generalization of the Stein-Tomas-Mockenhaupt Restriction Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15267 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension Polasek, Iván Rela, Ezequiel Classical Analysis and ODEs 42A38, 28A80 In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This builds upon Mockenhaupt's Restriction Theorem; while this theorem could only be applied to sets of positive Fourier dimension, we show that the existence of a measure with polylogarithmic Fourier decay combined with full Hausdorff dimension 1 on the real line is enough to guarantee restriction. In order to achieve this, we combine two different tools: a modification of a construction from a recent work of Li and Liu to produce a set with specific Hausdorff and Fourier dimensions, and a generalization of the Stein-Tomas-Mockenhaupt Restriction Theorem. |
| title | Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension |
| topic | Classical Analysis and ODEs 42A38, 28A80 |
| url | https://arxiv.org/abs/2602.15267 |