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SUMMARY:Benjamin Baily (Williams College)
DTSTART:20220524T183000Z
DTEND:20220524T185500Z
DTSTAMP:20260423T011340Z
UID:CANT2022/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/14/
 ">Large sets are sumsets</a>\nby Benjamin Baily (Williams College) as part
  of Combinatorial and additive number theory (CANT 2022)\n\n\nAbstract\nLe
 t $[n] :=\\{0\,1\,2\,\\dots\,n\\}$. Intuitively\, all large subsets of $[n
 ]$ have additive structure\, and Roth famously made this precise by\nfindi
 ng constants $c$\, $N > 0$ such that for $n \\geq N$\, any subset of $[n]$
  containing more than $\\frac{cn}{\\log\\log n}$ elements must contain an\
 narithmetic progression of length $3$. We establish a different interpreta
 tion of the intuition by finding explicit constants $\\alpha = \\frac{1}{\
 \log\n2}$ and $\\beta = \\frac{1}{\\log 1.325}$ such that\, for sufficient
 ly large $n$\, we have:\n\\begin{enumerate}\n\\item[(i)] any subset of $[n
 ]$ with more than $n-\\alpha \\log n$ elements has a nontrivial decomposit
 ion as the sum of two sets\, and\n\n\\item [(ii)]there exists a subset of 
 $[n]$ of size $n - \\beta \\log n$ at least  that has no such decompositio
 n.\n\n\\end{enumerate} We also prove\, using these methods\, a higher-dime
 nsional\nanalogue of results (i) and (ii). Notably\, our threshold at whic
 h\nstructure appears is far higher than Roth's.\n\nThis work was joint wit
 h Justine Dell\, Sophia Dever\, Adam Dionne\, Faye\nJackson\, Leo Goldmakh
 er\, Gal Gross\, Steven J. Miller\, Ethan Pesikoff\, Huy\nTuan Pham\, Luke
  Reifenberg\, and Vidya Venkatesh.     \\\\\n
LOCATION:https://researchseminars.org/talk/CANT2022/14/
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