VC-dimension of Salem sets over finite fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917096823193600 |
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| author | Diallo, Moustapha McDonald, Brian |
| author_facet | Diallo, Moustapha McDonald, Brian |
| contents | The VC-dimension, introduced by Vapnik and Chervonenkis in 1968 in the context of learning theory, has in recent years provided a rich source of problems in combinatorial geometry. Given $E\subseteq \mathbb{F}_q^d$ or $E\subseteq \mathbb{R}^d$, finding lower bounds on the VC-dimension of hypothesis classes defined by geometric objects such as spheres and hyperplanes is equivalent to constructing appropriate geometric configurations in $E$. The complexity of these configurations increases exponentially with the VC-dimension.
These questions are related to the Erdős distance problem and the Falconer problem when considering a hypothesis class defined by spheres. In particular, the Erdős distance problem over finite fields is equivalent to showing that the VC-dimension of translates of a sphere of radius $t$ is at least one for all nonzero $t\in \mathbb{F}_q$. In this paper, we show that many of the existing techniques for distance problems over finite fields can be extended to a much broader context, not relying on the specific geometry of circles and spheres. We provide a unified framework which allows us to simultaneously study highly structured sets such as algebraic curves, as well as random sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08963 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | VC-dimension of Salem sets over finite fields Diallo, Moustapha McDonald, Brian Combinatorics Classical Analysis and ODEs Number Theory The VC-dimension, introduced by Vapnik and Chervonenkis in 1968 in the context of learning theory, has in recent years provided a rich source of problems in combinatorial geometry. Given $E\subseteq \mathbb{F}_q^d$ or $E\subseteq \mathbb{R}^d$, finding lower bounds on the VC-dimension of hypothesis classes defined by geometric objects such as spheres and hyperplanes is equivalent to constructing appropriate geometric configurations in $E$. The complexity of these configurations increases exponentially with the VC-dimension. These questions are related to the Erdős distance problem and the Falconer problem when considering a hypothesis class defined by spheres. In particular, the Erdős distance problem over finite fields is equivalent to showing that the VC-dimension of translates of a sphere of radius $t$ is at least one for all nonzero $t\in \mathbb{F}_q$. In this paper, we show that many of the existing techniques for distance problems over finite fields can be extended to a much broader context, not relying on the specific geometry of circles and spheres. We provide a unified framework which allows us to simultaneously study highly structured sets such as algebraic curves, as well as random sets. |
| title | VC-dimension of Salem sets over finite fields |
| topic | Combinatorics Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2511.08963 |