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SUMMARY:Ashwin Sah (MIT)
DTSTART:20210315T140000Z
DTEND:20210315T150000Z
DTSTAMP:20260423T021132Z
UID:EPC/51
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EPC/51/">Pop
 ular differences for matrix patterns</a>\nby Ashwin Sah (MIT) as part of E
 xtremal and probabilistic combinatorics webinar\n\n\nAbstract\nWe study ma
 trix patterns including those of the form $x\,x+M_1d\,x+M_2d\,x+(M_1+M_2)d
 $ in abelian groups $G^k$ for integer matrices $M_1\,M_2$. If $A\\subseteq
  G^k$ has density $\\alpha$\, one might expect based on recent conjectures
  of Ackelsberg\, Bergelson\, and Best that there is $d\\neq 0$ such that \
 \[\\#\\{x \\in G^k: x\, x+M_1d\, x+M_2d\, x+(M_1+M_2)d \\in A\\} \\ge (\\a
 lpha^4-o(1))|G|^k\\] as long as $M_1\,M_2\,M_1\\pm M_2$ define automorphis
 ms of $G^k$. We show this conjecture holds in $G = \\mathbb{F}_p^n$ for od
 d $p$ given an additional spectral condition\, but is false without this c
 ondition. Explicitly\, we show the <i>rotated squares</i> pattern is false
  over $\\mathbb{F}_5^n$. This is in surprising contrast to the theory of p
 opular differences of one-dimensional patterns.\n
LOCATION:https://researchseminars.org/talk/EPC/51/
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