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SUMMARY:Ilya Shkredov (Purdue University)
DTSTART:20260713T180000Z
DTEND:20260713T185000Z
DTSTAMP:20260710T111702Z
UID:CANT2026/17
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/17/
 ">On Korobov's optimal coefficients</a>\nby Ilya Shkredov (Purdue Universi
 ty) as part of Combinatorial and additive number theory seminar (CANT 2026
 )\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor
 ).\n\nAbstract\nLet $p$ be a prime number\, $d$ be a positive integer\, an
 d $M\\ge 1$ be a real parameter. A tuple $(a_1\,\\dots\, a_d) \\in \\mathb
 f{F}^d_p$ is called a tuple of (Korobov) {\\it optimal coefficients} if\, 
 for any nonzero $x\\in \\mathbf{F}_p$\, the inequality$$\n	x|a_1 x| \\dots
  |a_d x| \\ge \\frac{p^d}{M} \n$$  holds. \n	These famous coefficients ari
 se naturally in numerical integration problems. 	Namely\, if a tuple $(a_1
 \, \\dots\, a_d)$ satisfying the inequality is found\, then any function $
 f:[0\,1]^d \\to \\mathbf{R}$ can be integrated using the formula $$\n\\lef
 t| \\int_{[0\,1]^d} f(x)\\\,dx - \\frac{1}{p} \\sum_{x=1}^{p} f\\left(\\fr
 ac{a_1 x}{p}\, \\dots\, \\frac{a_d x}{p} \\right) \\right| \\ll \\frac{M\\
 cdot \\mathrm{V}(f)}{p} \\\,\,\n$$\n where $\\mathrm{V}(f)$ is the Hardy--
 Krause variation of the function $f$. \nKorobov proved that the case $M=O(
 (\\log p)^{d-1})$ is always realizable\, whereas the special case $d=1$\, 
 $M=O(1)$ is equivalent to the well-known Zaremba conjecture.\nFor $d>1$ an
 d arbitrary $M$\, only a few results are known. In our talk\, we will prov
 ide an overview of the problems in this area and describe recent advances 
 and connections to other topics in number theory.\n
LOCATION:https://researchseminars.org/talk/CANT2026/17/
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