Improved bounds for five-term arithmetic progressions

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 $U^4$-inv...

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Hauptverfasser:	Leng, James, Sah, Ashwin, Sawhney, Mehtaab
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Schlagworte:	Mathematics - Combinatorics Mathematics - Number Theory
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creator	Leng, James Sah, Ashwin Sawhney, Mehtaab
description	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.
doi_str_mv	10.48550/arxiv.2312.10776
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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.</abstract><doi>10.48550/arxiv.2312.10776</doi><oa>free_for_read</oa></addata></record>
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title	Improved bounds for five-term arithmetic progressions
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