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Tue, Oct 06, 2020
Number Theory Seminar 2:00 PM Online (Zoom) New Results on Polynomial Values in Difference Sets Alex Rice, Millsaps College Host: John Doyle E-mail Paul Fili (paul.fili@okstate.edu) for the Zoom link. Abstract: It is a well known result in arithmetic combinatorics, established independently by Sárközy and Furstenberg, that if a set of integers has positive upper density (in other words the proportion of integers contained in the set does NOT limit to zero), then the set must contain two distinct elements that differ by a perfect square. The best-known quantitative bounds for this result were established with an intricate Fourier analytic argument by Pintz, Steiger, and Szemerédi. In this talk, we discuss the extension of these bounds from perfect squares to the largest possible class of univariate polynomials, as well as even better bounds for a large class of multivariate polynomials. In both settings we utilize a polynomial-specific sieve as a bridge to optimal exponential sum estimates (due to Weil and Deligne) over finite fields, and in the multivariate setting we invoke a variety of additional tools from algebraic geometry. This includes joint work with John Doyle.
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