Harmonic Analysis
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Combinatorial complexity of convex sequences. Average decay of the Fourier transform and applications
speaker: Alex Iosevich (University of Missouri)abstract: 1: Let ${\{bj\}}{j=1}N$ be a convex sequence of real numbers. Let ${\Cal N}d$ denote the number of solutions of the equtions $$ b{j1}+\dots+b{jd}=b{j'1}+\dots+b{j'd} \tag$$ We will show that ${\Cal N}d \lesssim N{2d-2+2{-d+1}}$ without any additional assumptions on the sequence $bk$. This is joint work with M. Rudnev and V. Ten. The result was independently obtained by S. Konyagin using different methods.
2: We shall discuss some Fourier analytic inequalities related to the average decay of the Fourier transform of compactly supported measures, and their applications to problems in geometric combinatorics.
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