Polynomial Wolff axioms and Kakeya-type estimates in R4
We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ-tubes satisfies the Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a plane. A collection of tubes satisfies the polynomial W...
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Oxford University Press - London Mathematical Society
2018
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| Online Access: | http://hdl.handle.net/1721.1/115562 https://orcid.org/0000-0002-1302-8657 https://orcid.org/0000-0001-5129-8300 |
| _version_ | 1826214725358190592 |
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| author | Guth, Lawrence Zahl, Joshua |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Guth, Lawrence Zahl, Joshua |
| author_sort | Guth, Lawrence |
| collection | MIT |
| description | We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ-tubes satisfies the Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a low degree algebraic variety. First, we prove that if a set of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least δ1-1/28. We also prove a more technical statement which is analogous to a maximal function estimate at dimension 3+1/28. Second, we prove that if a collection of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least δ3/4. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension 3+1/4. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension 3+1/28, and in particular this implies that every Kakeya set in R4 must have Hausdorff dimension at least 3+1/28. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995. |
| first_indexed | 2024-09-23T16:10:10Z |
| format | Article |
| id | mit-1721.1/115562 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T16:10:10Z |
| publishDate | 2018 |
| publisher | Oxford University Press - London Mathematical Society |
| record_format | dspace |
| spelling | mit-1721.1/1155622022-10-02T06:50:26Z Polynomial Wolff axioms and Kakeya-type estimates in R4 Guth, Lawrence Zahl, Joshua Massachusetts Institute of Technology. Department of Mathematics Guth, Lawrence Zahl, Joshua We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ-tubes satisfies the Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a low degree algebraic variety. First, we prove that if a set of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least δ1-1/28. We also prove a more technical statement which is analogous to a maximal function estimate at dimension 3+1/28. Second, we prove that if a collection of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least δ3/4. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension 3+1/4. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension 3+1/28, and in particular this implies that every Kakeya set in R4 must have Hausdorff dimension at least 3+1/28. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995. 2018-05-22T18:09:48Z 2018-05-22T18:09:48Z 2018-04 2018-05-22T15:29:58Z Article http://purl.org/eprint/type/JournalArticle 0024-6115 1234-5678 http://hdl.handle.net/1721.1/115562 Guth, Larry, and Joshua Zahl. “Polynomial Wolff Axioms and Kakeya-Type Estimates in R4.” Proceedings of the London Mathematical Society, Apr. 2018. https://orcid.org/0000-0002-1302-8657 https://orcid.org/0000-0001-5129-8300 http://dx.doi.org/10.1112/plms.12138 Proceedings of the London Mathematical Society Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press - London Mathematical Society arXiv |
| spellingShingle | Guth, Lawrence Zahl, Joshua Polynomial Wolff axioms and Kakeya-type estimates in R4 |
| title | Polynomial Wolff axioms and Kakeya-type estimates in R4 |
| title_full | Polynomial Wolff axioms and Kakeya-type estimates in R4 |
| title_fullStr | Polynomial Wolff axioms and Kakeya-type estimates in R4 |
| title_full_unstemmed | Polynomial Wolff axioms and Kakeya-type estimates in R4 |
| title_short | Polynomial Wolff axioms and Kakeya-type estimates in R4 |
| title_sort | polynomial wolff axioms and kakeya type estimates in r4 |
| url | http://hdl.handle.net/1721.1/115562 https://orcid.org/0000-0002-1302-8657 https://orcid.org/0000-0001-5129-8300 |
| work_keys_str_mv | AT guthlawrence polynomialwolffaxiomsandkakeyatypeestimatesinr4 AT zahljoshua polynomialwolffaxiomsandkakeyatypeestimatesinr4 |