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SUMMARY:Firdavs Rakhmonov (University of St. Andrews\, UK)
DTSTART:20250523T160000Z
DTEND:20250523T162500Z
DTSTAMP:20260423T024834Z
UID:CANT2025/64
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/64/
 ">Exceptional projections in finite fields: Fourier analytic bounds and in
 cidence geometry</a>\nby Firdavs Rakhmonov (University of St. Andrews\, UK
 ) as part of Combinatorial and additive number theory (CANT 2025)\n\nLectu
 re held in CUNY Graduate Center - Science Center (4th floor).\n\nAbstract\
 nWe consider the problem of bounding the number of exceptional projections
  (projections which are smaller than typical)  of a subset of a vector spa
 ce over a finite field.  We establish bounds that depend on $L^p$ estimate
 s for the Fourier transform\, improving various known bounds for sets with
  sufficiently good Fourier analytic properties.  The special case $p=2$ re
 covers a recent result of Bright and Gan (following Chen)\, which establis
 hed the finite field analogue of Peres--Schlag's bounds from the continuou
 s setting.\\\\\nWe prove several auxiliary results of independent interest
 \, including a character sum identity for subspaces (solving a problem of 
 Chen)\, and an analogue of Plancherel's theorem for subspaces. These auxil
 iary results also have applications in affine incidence geometry\, that is
 \, the problem of estimating the number of incidences between a set of poi
 nts and a set of affine $k$-planes. We present a novel and direct proof of
  a well-known result in this area that avoids the use of spectral graph th
 eory\, and we provide simple examples demonstrating that these estimates a
 re sharp up to constants.  \\\\\nThis is joint work with Jonathan Fraser.\
 n
LOCATION:https://researchseminars.org/talk/CANT2025/64/
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