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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.08589 |
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| _version_ | 1866913164742885376 |
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| author | Fraser, Jonathan M. |
| author_facet | Fraser, Jonathan M. |
| contents | The Fourier transform plays a central role in many geometric and combinatorial problems cast in vector spaces over finite fields. If a set admits optimal $L^\infty$ bounds on its Fourier transform (that is, it is a Salem set), then it can often be analysed more easily. However, in many cases obtaining good \emph{uniform} bounds is not possible, even if `most' points admit good pointwise bounds. Motivated by this, we propose a framework where one systematically studies the $L^p$ averages of the Fourier transform and keeps track of how good the $L^p$ bounds are as a function of $p$. This captures more nuanced information about a set than, for example, asking whether it is Salem or not. We explore this idea by considering several examples and find that a rich theory emerges. Further, we provide various applications of this approach; including to sumset type problems, the finite fields distance conjecture, and the problem of counting $k$-simplices inside a given set. Our typical application is of the form:~if a set admits good $L^p$ bounds on its Fourier transform, then we are able to make strong geometric conclusions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$ averages of the Fourier transform in finite fields Fraser, Jonathan M. Combinatorics Classical Analysis and ODEs primary: 52C10, 43A25, secondary: 11B30, 52C35, 43A46, 11L40 The Fourier transform plays a central role in many geometric and combinatorial problems cast in vector spaces over finite fields. If a set admits optimal $L^\infty$ bounds on its Fourier transform (that is, it is a Salem set), then it can often be analysed more easily. However, in many cases obtaining good \emph{uniform} bounds is not possible, even if `most' points admit good pointwise bounds. Motivated by this, we propose a framework where one systematically studies the $L^p$ averages of the Fourier transform and keeps track of how good the $L^p$ bounds are as a function of $p$. This captures more nuanced information about a set than, for example, asking whether it is Salem or not. We explore this idea by considering several examples and find that a rich theory emerges. Further, we provide various applications of this approach; including to sumset type problems, the finite fields distance conjecture, and the problem of counting $k$-simplices inside a given set. Our typical application is of the form:~if a set admits good $L^p$ bounds on its Fourier transform, then we are able to make strong geometric conclusions. |
| title | $L^p$ averages of the Fourier transform in finite fields |
| topic | Combinatorics Classical Analysis and ODEs primary: 52C10, 43A25, secondary: 11B30, 52C35, 43A46, 11L40 |
| url | https://arxiv.org/abs/2407.08589 |