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.