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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.15135 |
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| _version_ | 1866911276182011904 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | The dimension of Kakeya sets can be bounded using sum-difference exponents $\SD(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions, asserts that the infimum of such exponents is $1$. The best upper bound on this infimum currently is $1.67513\dots$. In this note, inspired by numerical explorations from the tool \texttt{AlphaEvolve}, we study the regime where the cardinality of the set of slopes $R$ is bounded. In this regime, we establish that these exponents converge to $2$ at a rate controlled by the \emph{rational complexity} of $s$ relative to $R$, which measures how efficiently $s$ can be expressed as a rational combination of slopes in $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum-difference exponents for boundedly many slopes, and rational complexity Tao, Terence Combinatorics 11B30, 94A17 The dimension of Kakeya sets can be bounded using sum-difference exponents $\SD(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions, asserts that the infimum of such exponents is $1$. The best upper bound on this infimum currently is $1.67513\dots$. In this note, inspired by numerical explorations from the tool \texttt{AlphaEvolve}, we study the regime where the cardinality of the set of slopes $R$ is bounded. In this regime, we establish that these exponents converge to $2$ at a rate controlled by the \emph{rational complexity} of $s$ relative to $R$, which measures how efficiently $s$ can be expressed as a rational combination of slopes in $R$. |
| title | Sum-difference exponents for boundedly many slopes, and rational complexity |
| topic | Combinatorics 11B30, 94A17 |
| url | https://arxiv.org/abs/2511.15135 |