Analysis Seminar, Dept. of Mathematics and Applied Math. -- Σεμινάριο Ανάλυσης

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Analysis Seminar in Crete (2023-24)

Abstract: I will discuss tilings of the real line by translates of a function $f$, that is, systems $\{f(x - \lambda), \lambda \in \Lambda\}$ of translates of $f$ that form a partition of unity. Which functions $f$ can tile by translations, and what can be the structure of the translation set $\Lambda$? I will survey the subject and present some recent results.

Abstract: Extremal properties of Littlewood polynomials (with coefficients $\pm 1$) have been extensively studied throughout the past century. Among special classes of Littlewood polynomials, particular attention has been given to so-called "Fekete polynomials" (with coefficients being Legendre symbols). Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials and their extremal properties have attracted considerable attention, particularly due to their intimate connection with the putative Siegel zero and the small class number problem. The goal of this talk is to discuss a general approach to understanding the behaviour of such polynomials, resolving several open problems. This is based on joint work with Y. Lamzouri and M. Munsch.

Abstract: A set W in the plane is unit-avoiding if no two points of W are unit distance away. Erdős conjectured that the maximal upper density of such a set is less than 1/4. In a recent joint work with G. Ambrus, A. Csiszarik, D. Varga, P. Zsamboki, we have proved this conjecture. The proof is an interesting mix of Fourier analysis, geometry, combinatorics and computer science.