Combinatorial complexity of convex sequences. Average decay of the Fourier transform and applications

speaker: Alex Iosevich (University of Missouri)

abstract: 1: Let ${\{b_j\}}_{j=1}^N$ be a convex sequence of real numbers. Let ${\Cal N}_d$ denote the number of solutions of the equtions $$ b_{j_1}+\dots+b_{j_d}=b_{j'_1}+\dots+b_{j'_d} \tag$$ We will show that ${\Cal N}_d \lesssim N^{2d-2+2^{-d+1}}$ without any additional assumptions on the sequence $b_k$. 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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