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.

We discuss convergence (in $L^2$) results for multiple ergodic averages along sequences of polynomial growth evaluated at primes. Building on the work of Frantzikinakis, Host, and Kra who showed that polynomial ergodic averages along primes converge, we generalize their results to other sequences with polynomial growth. Combining our results with Furstenberg's correspondence principle, we derive several applications in combinatorics. The most interesting application is that positive density subsets of $\mathbb{N}$ contain arbitrarily long arithmetic progressions with common difference of the form $\lfloor{p^c}\rfloor$, where $c$ is a positive non-integer and $p$ is a prime number. The main tools in the proof are a recent result of Matomäki, Shao, Tao, and Teräväinen on the uniformity of the von Mangoldt function in short intervals, a polynomial approximation of our sequences with good equidistribution properties, and a lifting trick that allows us to replace $\mathbb{Z}$-actions on a probability space by $\mathbb{R}$-actions on an extension of the original system.

Abstract: An algebraic equation is partition regular if every finite coloring of the integers has monochromatic solutions. Necessary and sufficient conditions for partition regularity of linear equations were given by Radó in 1933. However, very little is currently known about homogeneous quadratic equations in three variables; the most notable problem is the partition regularity of Pythagorean triples. We will prove that Pythagorean pairs are partition regular, i.e., for any finite coloring of the integers we can find $x, y$ of the same color such that $x^2+y^2=z^2$ for some integer $z$ (and a similar statement with the roles of $y$ and $z$ swapped). Our method combines Gowers uniformity properties of bounded multiplicative functions and a new approach based on concentration estimates of pretentious multiplicative functions.

Abstract: Recently Greenfeld and Tao found an example of a finite subset in $\mathbb{Z}^d$ (for some large $d$) which tiles $\mathbb{Z}^d$ by translations but only aperiodically, thus disproving the so-called Periodic Tiling Conjecture in high enough dimension.

Roughly 20 years ago the Fuglede (or Spectral set) conjecture was disproved by Tao (in the spectral implies tiling direction) and by Kolountzakis and Matolcsi (in the tiling implies spectral direction). In this problem the dimension $d$ eventually got down to 3 for both directions.

In both these problems (aperiodicity and Fuglede conjecture) the examples found are highly dispersed subsets of $\mathbb{Z}^d$. In this work we show how to modify these examples to obtain (pathwise) connected subsets of $\mathbb{Z}^d$ as examples by increasing the dimension $d$.

Roth's theorem and the Hardy-Littlewood majorant problem for thin subsets of Primes

Abstract: We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy-Littlewood majorant property holds for these sets. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to 1, which are primes of the form $\lfloor n^c\rfloor$ for a fixed $c>1$.

Abstract: We investigate isoperimetric type problems restricted to some classes of parallelotopes. Discrete and continuous versions are considered.

Thu, 28 Mar 2024, 11:15, Room: Α303
Speaker: Vassili Nestoridis (University of Athens)

Abstract: In Mergelyan type theorems we uniformly approximate functions in compact sets $K$ by polynomials, rational functions or holomorphic functions in varying open sets containing $K$. We improve uniform approximation in $K$ by approximating as well all derivatives. The case of one complex variable is contained in the article arXiv:2006.02389 by Armeniakos-Kotsovolis-Nestoridis, published in Monatchefte fur Mathematik (2022). In the present talk, we will present generalizations for several complex variables, based on a collaboration with P. M. Gauthier.

Abstract: We will discuss inequalities that compare the volume of a set with the volume of lower dimensional sections of the set. Such inequalities can be derived from Brascamp-Lieb and its reverse.

Abstract: We are going to discuss some simple connections between the uncertainty principle and the exact recovery of functions from incomplete data. The role of Fourier restriction inequalities will play an important role.

Abstract: The Liouville function $\lambda$ is the arithmetic function for which $\lambda(1)=1$, $\lambda(n)=-1$ when $n$ is the product of an odd number of primes, and $\lambda(n)=1$ when $n$ is the product of an even number of primes. The prime number theorem implies that $$\sum_{n\leqslant x}\lambda(n)=o(x),\quad \text{as} \quad x \to \infty,$$ which means that the sign of $\lambda(n)$ changes frequently as $n$ grows. Chowla expected a more general version of this asymptotic to hold, and in 1965, he conjectured that for any fixed distinct non-negative integers $h_1,\ldots,h_k$, we should have that $$\sum_{n\leqslant x}\lambda(n+h_1)\cdots\lambda(n+h_k)=o(x)\quad \text{as} \quad x \to \infty.$$ An unconditional answer to this conjecture is yet to be found, and in this talk, we are taking a conditional approach towards it. In particular, we will discuss Chowla's conjecture under the existence of Siegel zeroes. We will start by introducing the notion of a Siegel zero and then we will heuristically explain why the existence of these zeroes is a useful assumption when "attacking" Chowla's conjecture. We will continue by describing how one can establish a bound for the sums $\sum_{n\leqslant x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ using the presence of Siegel zeroes. The estimate that we will present constitutes an improvement over the previous related results of Germán and Kátai, Chinis, and Tao and Teräväinen. This talk is based on on-going joint work with Mikko Jaskari.