BGU Probability and Ergodic Theory (PET) seminar Ical Atom

The covering radius of a shift space is a quantity of interest for information-theoretic applications of data transmission over noisy channels. In this talk we will explain what is the covering radius of a sofic shift and why people care about it. We will outline a proof that the covering radius of a primitive sofic shift is always a rational number, and outline an algorithm to compute the covering radius from a labeled graph presentation. We will also briefly explain how these results relate to dynamics, to a certain zero-sum two-player game and to an old meta-conjecture about typical ground states in statistical mechanics. The notions will be defined, no specific background assumed. Based on joint work with Aidan Young as in https://arxiv.org/abs/2603.21449, and previous joint work with Dor Elimelech and Moshe Schwartz as in https://ieeexplore.ieee.org/document/10360152

For a minimal action of a countable group G on a compact space X, we establish necessary conditions for the simplicity of the corresponding reduced crossed product C*-algebras in terms of stabilizer subgroups. In particular, our result gives a complete characterization of the simplicity of the reduced crossed product associated with minimal actions of linear groups, answering a question of Ozawa (2014) for these groups. Joint work with Mehrdad Kalantar

In 1965, Wolfgang Schmidt introduced the $(\alpha,\beta)$-Schmidt game as a dynamical tool for studying fundamental sets in Diophantine approximation. In particular, he proved that in Hilbert spaces these games can be used to obtain lower bounds on the Hausdorff dimension of sets that are small from the measure-theoretic point of view but large in a fractal sense. Schmidt’s approach relies on the underlying geometry of the space.

In this talk, I will introduce these games and present an analogous result in the setting of complete doubling metric spaces, where we replace geometric arguments with a purely game-theoretic approach.

This is joint work with Itamar Bellaïche. No prior knowledge of game theory is assumed.

Let $k \geq 2$ be an integer. In 2000, Baake, Moody, and Pleasants proved that the set of lattice points in $\mathbb{Z}^k$ visible from the origin has pure point diffraction. It is also known that irreducible cut-and-project sets—such as the Ammann-Beenker point set—exhibit pure point diffraction.

Let $S$ be a finite subset of $\mathbb{Z}^k$, and let $V(S)$ be the set of points simultaneously visible from $S$. We will discuss the diffraction spectrum of the set $V(S)$ and the diffraction spectrum of the set of visibility from the origin in certain classes of irreducible cut-and-project sets. Joint work with Carlos Ospina.

The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak says that the L^2 mass of eigenfunctions of the Laplacian in hyperbolic manifolds equidistributes, as the eigenvalues tend to infinity. We consider a special class of such functions, Hecke—Maass forms, that are central in number theory. The conjecture has been established for these functions in dimension 2 and 3, but in dimension 4 there is a new challenge: one needs to rule out concentration of measure along certain large totally geodesic submanifolds. We will discuss our recent result in which we overcome this difficulty for a particular sequence of eigenfunctions known in number theory as lifts. This is a joint work with Alexandre de Faveri.