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SUMMARY:Improved lower bounds for Szemeredi’s theorem.  - Zach Hunter (E
 TH Zurich) 
DTSTART:20231019T133000Z
DTEND:20231019T143000Z
UID:TALK207487@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:Let $r_k(N)$ denote cardinality of the largest subset of $\\{1
 \,…\,N\\}$ which does not contain an arithmetic progression of length $k
 $. Since 1961\, the best lower bound (up to lower order terms) has been du
 e to Rankin\, establishing $r_k(N) \\ge N exp(-(c_k+o(1)) \\log^{p_k}(N))$
  for certain explicit constants $c_k\,p_k> 0$\, generalizing a constructio
 n of Behrend.\n\nWe shall establish new bounds for this problem\, improvin
 g the constant $c_k$ for all $k\\ge 7$ (with the same value of $p_k$). Our
  methods also have implications for related problems in finite fields. 
LOCATION:MR12
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