Degree reduction and graininess for Kakeya-type sets in R[superscript 3]
Let T be a set of cylindrical tubes in ℝ[superscrpit 3] of length N and radius 1. If the union of the tubes has volume N[superscript 3-σ], and each point in the union lies in tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale N[superscript...
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| Format: | Article |
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European Mathematical Publishing House
2018
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| Online Access: | http://hdl.handle.net/1721.1/115569 https://orcid.org/0000-0002-1302-8657 |
| _version_ | 1826197532053602304 |
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| author | Guth, Lawrence |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Guth, Lawrence |
| author_sort | Guth, Lawrence |
| collection | MIT |
| description | Let T be a set of cylindrical tubes in ℝ[superscrpit 3] of length N and radius 1. If the union of the tubes has volume N[superscript 3-σ], and each point in the union lies in tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale N[superscript σ ], the tubes are clustered into rectangular slabs of dimension 1 x N[superscript σ] x N[superscript σ]. This estimate generalizes the graininess estimate in [7]. The proof is based on modeling the union of tubes with a high-degree polynomial. Keywords: Kakeya set, incidence geometry, polynomial method |
| first_indexed | 2024-09-23T10:49:06Z |
| format | Article |
| id | mit-1721.1/115569 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T10:49:06Z |
| publishDate | 2018 |
| publisher | European Mathematical Publishing House |
| record_format | dspace |
| spelling | mit-1721.1/1155692022-09-27T15:13:40Z Degree reduction and graininess for Kakeya-type sets in R[superscript 3] Guth, Lawrence Massachusetts Institute of Technology. Department of Mathematics Guth, Lawrence Let T be a set of cylindrical tubes in ℝ[superscrpit 3] of length N and radius 1. If the union of the tubes has volume N[superscript 3-σ], and each point in the union lies in tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale N[superscript σ ], the tubes are clustered into rectangular slabs of dimension 1 x N[superscript σ] x N[superscript σ]. This estimate generalizes the graininess estimate in [7]. The proof is based on modeling the union of tubes with a high-degree polynomial. Keywords: Kakeya set, incidence geometry, polynomial method 2018-05-22T19:26:36Z 2018-05-22T19:26:36Z 2016-06 2018-05-22T16:10:14Z Article http://purl.org/eprint/type/JournalArticle 0213-2230 http://hdl.handle.net/1721.1/115569 Guth, Larry. “Degree Reduction and Graininess for Kakeya-Type Sets in R[superscript 3].” Revista Matemática Iberoamericana, vol. 32, no. 2, 2016, pp. 447–94. https://orcid.org/0000-0002-1302-8657 http://dx.doi.org/10.4171/RMI/891 Revista Matemática Iberoamericana Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf European Mathematical Publishing House arXiv |
| spellingShingle | Guth, Lawrence Degree reduction and graininess for Kakeya-type sets in R[superscript 3] |
| title | Degree reduction and graininess for Kakeya-type sets in R[superscript 3] |
| title_full | Degree reduction and graininess for Kakeya-type sets in R[superscript 3] |
| title_fullStr | Degree reduction and graininess for Kakeya-type sets in R[superscript 3] |
| title_full_unstemmed | Degree reduction and graininess for Kakeya-type sets in R[superscript 3] |
| title_short | Degree reduction and graininess for Kakeya-type sets in R[superscript 3] |
| title_sort | degree reduction and graininess for kakeya type sets in r superscript 3 |
| url | http://hdl.handle.net/1721.1/115569 https://orcid.org/0000-0002-1302-8657 |
| work_keys_str_mv | AT guthlawrence degreereductionandgraininessforkakeyatypesetsinrsuperscript3 |