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Improved bounds for five-term arithmetic progressions
- Publication Year :
- 2023
-
Abstract
- Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work is a consequence of recent improved bounds on the $U^4$-inverse theorem of the first author and the fact that $3$-step nilsequences may be approximated by locally cubic functions on shifted Bohr sets. This combined with the density increment strategy of Heath-Brown and Szemer{\'e}di, codified by Green and Tao, gives the desired result.<br />Comment: 35 pages, comments welcome!
- Subjects :
- Mathematics - Number Theory
Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2312.10776
- Document Type :
- Working Paper