A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture
| dc.creator | Katz, Nets Hawk | |
| dc.creator | Tao, Terence | |
| dc.date | 1999-06-14 | |
| dc.date | 2000-01-20 | |
| dc.date.accessioned | 2026-07-07T05:29:31Z | |
| dc.date.available | 2026-07-07T05:29:31Z | |
| dc.description | Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain improved the trivial upper bound of $N^2$ to $N^{2-1/13}$, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to $N^{2-1/6}$, and obtain the further improvement of $N^{2-1/4}$ if we also know that $# \{a+2b: (a,b) \in G\} \leq N$. We conclude that Besicovitch sets in $\R^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski dimension at least $4n/7 + 3/7$. This is new for $n > 8$. | |
| dc.description | 6 pages, submitted to Math Research Letters; improved bounds in revised version; typoes corrected in second revised version | |
| dc.identifier | https://arxiv.org/abs/math/9906097 | |
| dc.identifier | http://arxiv.org/abs/math/9906097 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78665 | |
| dc.subject | Combinatorics | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 42B25, 05C35 | |
| dc.title | A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture | |
| dc.type | text |