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

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Analysis Seminar in Crete (2025-26)

Abstract: In 1974 Hindman proved his well-known finite sums theorem, showing that IP sets are partition regular. We survey some recent combinatorial results surrounding the problem of extending this result to a density analogue. We also review the newly developed ergodic theoretic machinery that has been developed and exploited in order to obtain these results.

Abstract: Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture can be reduced to sets which tile a finite cyclic group. In this talk we consider a natural relaxation of this problem, where we replace sets with nonnegative functions, such that $f*g$ is a functional tiling, and satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.

Abstract: We prove that for any two lattices $L, M \subseteq \RR^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \RR^d$ such that $E$ tiles $\RR^d$ when translated by $L$ or by $M$. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\RR^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$.

Abstract: Informally, a real number is normal in base 10 if each of the digits 0, 1, . . . , 9 shows up with frequency 1/10 in its decimal expansion, each pair of digits 00, 01, . . . , 99 shows up with frequency 1/100 in its decimal expansion and so on.

There are many basic and easily stated (but very difficult!) open questions revolving around normality. For example, even determining the normality of $\pi$ in any base appears to be far out of reach of modern mathematics.

Depending on the interest of the audience, we may explore connections between normal numbers and other areas of math such as ergodic theory, descriptive set theory, computability theory, probability theory, and others.

Abstract: We provide sharp bounds for moments of sums of exponential random variables and their connection with Hunter's conjecture.

Abstract: We study the spectrality of arc-length measures supported on the union of two line segments in the plane. We show that any such spectral measure must admit a line spectrum. Moreover, when the two segments are non-parallel, such spectral measure admits only line spectra. Thus, in this case every spectrum is one dimensional. In addition we show that this property fails for unions of three or more segments in the plane. We construct some arc-length spectral measures supported on the union of at least three line segments such that none of its spectra is contained in a line. Finally, we work in the general framework of arc-length measures supported on finite unions of curves in $\mathbb{R}^d$. We show that the size of any orthogonal set for such a measure inside a ball of radius $R$ grows at most linearly in $R$. We also give an alternative proof of this bound, and in fact obtain a more general result of growth rate of orthogonal sets for Ahlfors--David regular measures in $\mathbb{R}^d$ (not restricted to the one-dimensional setting).

Abstract: Caffarelli–Kohn–Nirenberg (CKN) weighted interpolation inequalities are fundamental tools in analysis of partial differential equations, geometric analysis and related fields for establishing relationships between functions and their gradients. Such types of inequalities have been extensively studied in several geometric and functional analytic settings. In this talk, I will present certain classes of Caffarelli-Kohn-Nirenberg (CKN) inequalities on general homogeneous Lie groups with no additional structure. We will follow a straightforward approach based on symmetrization arguments. We consider CKN inequalities with optimal constants and determination of extremals. Also we establish sharp improvements of weighted Hardy inequalities on general homogeneous Lie groups, considering some critical (limiting) cases of the parameters as well. Furthermore, we derive sharp weighted Hardy-Sobolev (HS) type inequalities of a new scale-invariant form, and provide alternative proofs in the critical cases. Even in the Euclidean setting (Abelian groups), the results imply new sharp CKN inequalities in the case of arbitrary homogeneous quasi-norms and dilation structures, and refined versions involving the radial derivative in place of the full gradient. Our results unify, improve, and extend several related estimates regarding weighted HS inequalities in both critical and non-critical cases.