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SUMMARY:Trevor Wooley (Purdue University)
DTSTART:20200604T153000Z
DTEND:20200604T155500Z
DTSTAMP:20260423T011158Z
UID:CANT2020/48
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/48/
 ">Condensation and densification for sets of large diameter</a>\nby Trevor
  Wooley (Purdue University) as part of Combinatorial and additive number t
 heory (CANT 2021)\n\n\nAbstract\nConsider a set of integers $A$ having fin
 ite diameter $X$\, so that\n\\[\n\\sup A-\\inf A=X<\\infty \,\n\\]\nand a 
 system of simultaneous polynomial equations $P_1(\\mathbf x)=\\ldots \n=P_
 r(\\mathbf x)=0$ to be solved with $\\mathbf x\\in A^s$. In many circumsta
 nces\, one can \nshow that the number $N(A\;\\mathbf P)$ of solutions of t
 his system satisfies \n$N(A\;\\mathbf P)\\ll X^\\epsilon |A|^\\theta$ for 
 a suitable $\\theta < s$ and any $\\epsilon>0$. \nSuch is the case with mo
 dern variants of Vinogradov's mean value theorem due to the \nauthor\, and
  likewise Bourgain\, Demeter and Guth. These estimates become worse than t
 rivial \nwhen the diameter $X$ is very large compared to $|A|$\, or equiva
 lently\, when the set $A$ is \nvery sparse. This motivates the problem of 
 seeking new sets of integers $A'$ in a certain \nsense ``isomorphic'' to $
 A$ having the property that (i) the diameter $X'$ of $A'$ is smaller \ntha
 n $X$\, and (ii) the set $A'$ preserves the salient features of the soluti
 on set of the \nsystem of equations $P_1(\\mathbf x)=\\ldots =P_r(\\mathbf
  x)=0$. We will report on our \nspeculative meditations (both results and 
 non-results) concerning this problem closely \nassociated with the topic o
 f Freiman homomorphisms.\n
LOCATION:https://researchseminars.org/talk/CANT2020/48/
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