ETIENE - Equidistribution in Number Theory

Hybrid seminar series

October 6, 2025, at 16:30 (Jussieu, room 15-25.502): Farrell Brumley
The mixing conjecture for more discriminants

The mixing conjecture of Michel--Venkatesh can be thought of as an ergodic theoretic refinement of the André--Oort conjecture in the case of a product of modular curves. There are essentially two known approaches to this conjecture, both of which have yielded only conditional results. The first, due to Khayutin in a landmark paper, uses measure classification of higher rank diagonalizable actions and analytic number theory, but imposes splitting conditions on the discriminants and assumes no Siegel zeros. The other approach, due to Blomer, myself, and Khayutin, uses automorphic forms and analytic number theory, and most notably requires the generalized Riemann hypothesis (GRH). I will survey these results and methods, and describe ongoing work, joint with Blomer and Radziwiłł, in which we weaken the appeal to GRH in a closely related problem on simultaneous equidistribution to a more accessible hypothesis on the abundance of small split primes. The latter condition, when quantified, is of comparable strength to the absence of Siegel zeros, and allows us to capture more discriminants in the mixing conjecture than was known previously.
September 29, 2025, at 16:30 (D 16.2, ETH): Emmanuel Kowalski
An application of equidistribution of exponential sums to graph theory (joint work with A. Forey, J. Fresán and Y. Wigderson)

We construct examples of deterministic families of graphs with semicircle spectral distribution and with other remarkable properties (known or expected). The main tool is the equidistribution theorem for discrete Fourier transforms of trace functions. In the application of this result, some further interesting arithmetic questions arise, including a variant of the Artin conjecture on primitive roots.
May 28, 2025, at 13:30 (ETH): René Pfitscher

Siegel transforms and counting rational approximations on flag varieties

In the divergence case of Khintchine’s theorem, Schmidt established an asymptotic formula for the number of rational approximations of bounded height to almost every real number. Using tools from homogeneous dynamics and the geometry of numbers, we prove a version of this theorem for intrinsic Diophantine approximation on projective quadrics, Grassmannians, and other examples of flag varieties.

Zoom link: https://ethz.zoom.us/j/68750218413
May 19, 2025, at 16:00 (Jussieu, room 15-25.502): Nicolas de Saxcé

Quantum ergodicity on compact homogeneous spaces

On a compact simple Lie group G with trivial center, we consider an averaging operator T on a finite symmetric set S of elements generating a dense subgroup Gamma. The statistics of the eigenvalues of T in a finite irreducible representation converge to a natural measure associated to the Cayley graph of (Gamma,S). One can then try to understand the eigenfunctions of T in such a representation, viewed as elements in L^2(G). The quantum ergodicity we show (following work of Brooks, Le Masson and Lindenstrauss on the 2-dimensional sphere) is that most of these eigenfunctions weakly converge to the Haar measure on G. The goal of the talk will be to show how both of the above statements -- on eigenvalues and eigenfunctions -- are strongly related to the fact that characters of compact simple Lie group concentrate near the identity, and to discuss generalizations of this phenomenon to relative characters that would allow us to prove quantum ergodicity on any quotient of the form G/H, for H a closed subgroup of G. (Work in progress with Emmanuel Schenck.)
May 5, 2025, at 16:00 (ETH): Théo Untrau

Wasserstein metrics and quantitative equidistribution of some exponential sums

In this talk, I will present an equidistribution result concerning a specific type of incomplete exponential sums over F_p that are defined by restricting the range of summation to the set of roots in F_p of a fixed polynomial with integer coefficients. We will see that the limit measure is related to the uniform measure on a closed subgroup of the torus which is not fully explicit in general. This leads to difficulties when one wants to obtain quantitative rates of equidistribution, as the usual box discrepancy is not adapted to the subtorus, and leads to technicalities when taking the pushforward measure to get a quantitative rate of convergence in the complex plane. I will explain how the use of metrics coming from the theory of optimal transportation can help resolving these issues.

This is part of an ongoing joint work with E. Kowalski.

The recording for this talk can be accessed here.
April 4, 2025, at 14:00 (Jussieu, room 15-16.411): Tuomas Sahlsten

Normal numbers, Fourier decay and additive chaos

Normal numbers are everywhere according to Lebesgue measure, but specific normal numbers with a given (e.g. Diophantine) property are very difficult to find. Via Weyl’s equidistribution criterion, studying Fourier transforms of measures on small fractal sets in reals is a fruitful tool to find normal numbers in small sets (e.g. finding normal numbers with bounded continued fraction expansions), and has partially contributed to various advances linked to harmonic analysis, random walks on groups, sum-product phenomenon in additive combinatorics and exponential mixing. In this talk I will overview what role Fourier decay has played in this area, outline how the discretised sum-product theorem appeared as a crucial tool in the example of limit sets of isometric group actions on hyperbolic plane and more general attractors to non-linear dynamical systems, and what recent challenges there are. If time permits, I will report on a recent joint work with Gaétan Leclerc and Sampo Paukkonen on Fourier decay in parabolic C^{1+alpha} systems that arose from this.