Ethan Ackelsberg

Equidistribution of orbits in 2-step nilpotent translational systems and applications

Solutions to several major problems in additive combinatorics rely on a dichotomy between “structure” and “randomness,” characterized in many cases by Gowers uniformity norms. In the ergodic theory context, there is a corresponding family of seminorms (the Host—Kra seminorms) producing Host—Kra factors as the “structure” behind combinatorial phenomena. A recent result of Jamneshan, Shalom, and Tao (2024) describes order 2 Host—Kra factors for actions of abelian groups as inverse limits of 2-step nilpotent translational systems defined on homogeneous spaces of 2-nilpotent locally compact Polish groups.

In this talk, we will discuss a new Ratner-type equidistribution theorem for orbits in 2-step nilpotent translational systems, generalizing previous equidistribution results for translations on (2-step) nilmanifolds. We will also discuss a combinatorial application of the equidistribution theorem to a problem about infinite triple sumsets in sets of positive density in abelian groups.