Generalized Arithmetic Kakeya
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909397398061056 |
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| author | Pohoata, Cosmin Zakharov, Dmitrii |
| author_facet | Pohoata, Cosmin Zakharov, Dmitrii |
| contents | Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as the `Arithmetic Kakeya Conjecture'. We consider a higher dimensional variant of this problem and prove an upper bound using a certain iterative argument. The main new ingredient in our proof is a general way to strengthen the sum-difference inequalities of Katz and Tao which might be of independent interest. As a corollary, we obtain a new lower bound for the Minkowski dimension of $(n, d)$-Besicovitch sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13395 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Arithmetic Kakeya Pohoata, Cosmin Zakharov, Dmitrii Combinatorics Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as the `Arithmetic Kakeya Conjecture'. We consider a higher dimensional variant of this problem and prove an upper bound using a certain iterative argument. The main new ingredient in our proof is a general way to strengthen the sum-difference inequalities of Katz and Tao which might be of independent interest. As a corollary, we obtain a new lower bound for the Minkowski dimension of $(n, d)$-Besicovitch sets. |
| title | Generalized Arithmetic Kakeya |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.13395 |