New Nikodym set constructions over finite fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915646259855360 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | For any fixed dimension $d \geq 3$ we construct a Nikodym set in $F_q^d$ of cardinality $q^d - (\frac{d-2}{\log 2} +1+o(1)) q^{d-1} \log q$ in the limit $q \to \infty$, when $q$ is an odd prime power. This improves upon the naive random construction, which gives a set of cardinality $q^d - (d-1+o(1)) q^{d-1} \log q$, and is new in the regime where $F_q$ has unbounded characteristic and $q$ not a perfect square. While the final proofs are completely human generated, the initial ideas of the construction were inspired by output from the tools \texttt{AlphaEvolve} and \texttt{DeepThink}. We also present a simple construction of Nikodym sets in $F_q^2$ for $q$ a perfect square that is a special case of known unital-based constructions, and matches the existing bounds of $q^2 - q^{3/2} + O(q \log q)$, assuming that $q$ is not the square of a prime $p \equiv 3 \pmod{4}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Nikodym set constructions over finite fields Tao, Terence Combinatorics 52C35, 05B25 For any fixed dimension $d \geq 3$ we construct a Nikodym set in $F_q^d$ of cardinality $q^d - (\frac{d-2}{\log 2} +1+o(1)) q^{d-1} \log q$ in the limit $q \to \infty$, when $q$ is an odd prime power. This improves upon the naive random construction, which gives a set of cardinality $q^d - (d-1+o(1)) q^{d-1} \log q$, and is new in the regime where $F_q$ has unbounded characteristic and $q$ not a perfect square. While the final proofs are completely human generated, the initial ideas of the construction were inspired by output from the tools \texttt{AlphaEvolve} and \texttt{DeepThink}. We also present a simple construction of Nikodym sets in $F_q^2$ for $q$ a perfect square that is a special case of known unital-based constructions, and matches the existing bounds of $q^2 - q^{3/2} + O(q \log q)$, assuming that $q$ is not the square of a prime $p \equiv 3 \pmod{4}$. |
| title | New Nikodym set constructions over finite fields |
| topic | Combinatorics 52C35, 05B25 |
| url | https://arxiv.org/abs/2511.07721 |