Activities This Week

Let X be an integral projective variety defined over Q of degree at least 2 and B > 0 an integer. The (uniform) dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown and Salberger, provides a uniform upper bound on the number of rational points of height at most B lying on X, where the bound depends only on the degree of X, the dimension of its ambient space, and on B.

In this talk, I will report on current developments which go beyond classical uniform dimension growth bounds, focusing on an affine variant (which implies the projective one).

This is based on joint work with Cluckers, Dèbes, Nguyen and Vermeulen.

We introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given topological dynamical systems $(X, T), (Y, S)$, and a continuous function $f \in C(X \times Y)$, what can be said about the extrema $\sup_{y \in Y} \inf_{x \in X} \lim_{k \to \infty} \frac{1}{k} \sum_{j = 0}^{k - 1} f \left( T^j x , S^j y \right)?$

We will present a refinement, developed in collaboration with Bjorklund, of Furstenberg’s ergodic-theoretic approach tailored to addressing the problem of identifying ‘twisted infinite patterns’ in positive-density subsets of the integer lattice. These patterns correspond to an infinite structure within the ‘sum-products’ formed by such sets. The talk is based on joint work with Bjorklund, Bulinski, and Skinner.