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Improved Bounds for Szemer\'{e}di's Theorem
- Publication Year :
- 2024
-
Abstract
- Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemer\'{e}di as reformulated by Green and Tao.<br />Comment: 13 pages
- Subjects :
- Mathematics - Combinatorics
Mathematics - Number Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2402.17995
- Document Type :
- Working Paper