Ergodic theory methods for the Sarnak Conjecture

The Ohio State University, April 25-26, 2020

We are planning a weekend workshop on recent developments around the Sarnak Conjecture. This would provide a forum for graduate students and other researchers to learn about recent developments in this field, to exchange ideas and discuss open problems. Everyone is welcome to participate.

Invited speakers

Nikos Frantzikinakis, University of Crete
Joanna Kułaga-Przymus, Nicolaus Copernicus University
Florian Richter, Northwestern University

We might also schedule a few short talks on ergodic theory/topological dynamics and interactions thereof with number theory.

Mini-course by Nikos Frantzikinakis, 4 lectures

Title: Ergodic properties of multiplicative functions and applications.

Abstract: The Möbius and the Liouville functions are multiplicative functions that encode important information related to distributional properties of the prime numbers. It is widely believed that their (non-zero) values are distributed randomly and various conjectures have been made based on this expectation. The Chowla conjecture (made in the 60's) asserts that the values of the Liouville function form a normal sequence of plus and minus ones, and the Möbius disjointness conjecture of Sarnak asserts that the Möbius function does not correlate with any bounded deterministic sequence. In recent years there has been important progress towards these conjectures by using a combination of techniques from analytic number theory and ergodic theory. My goal is to explain the basic principles behind these interactions and cover in some detail the proof of the Chowla conjecture on average, the odd and two point Chowla conjecture for logarithmic averages, and the logarithmic Sarnak conjecture for ergodic weights. Particular emphasis will be given on the ergodic theory aspects of these problems but no background in ergodic theory will be assumed.

Invited talks

Joanna Kułaga-Przymus, 2 lectures

Title: Where ergodic theory meets number theory: Sarnak’s conjecture.

Abstract: Sarnak’s conjecture is a prominent example of an interplay of two very rich and active areas: ergodic theory and number theory. Although Möbius orthogonality conjecture seems to be of topological nature, it turns out that, in fact, it has a strong ergodic theoretic flavour. To be more precise, one can ask (assuming zero Kolmogorov-Sinai entropy) which ergodic properties of a measure-theoretic dynamical system imply Möbius orthogonality of all its uniquely ergodic models. In particular, it turns out crucial that we understand joinings between (prime) powers: we aim for some kind of (asymptotic) disjointness. On the other hand, proving Möbius orthogonality even for periodic rotations requires understanding of number-theoretic properties of the Möbius function. The break-through here was made by Matomäki and Radziwiłł who studied the behaviour of non-pretentious multiplicative functions on short intervals. This, in turn, gave an impetus to study convergence on short intervals in ergodic theory. The main objective of the talk is to discuss these interactions, including recent developments and some resulting future directions.

Florian Richter

Title: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions.

Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak's conjecture, which deals with the disjointness of actions of (N,+) and (N,•). This talk is based on joint work with V. Bergelson.

Registration

Please register at least two weeks prior to the workshop so we have enough coffee and snacks for everyone. We expect to have some funds to support lodging and travel expenses for a limited number of junior participants. Only those participants who register by March 3rd will be considered for financial support. Supported participants will stay in shared rooms at Blackwell Hotel which is conveniently located within the walking distance from the math department. We will get back to you after March 3rd regarding lodging and the allowance available for travel cost reimbursement.

Travel

The Ohio State University is located in Columbus, Ohio. Columbus Airport (CMH, John Glenn Columbus International Airport) is the nearest airport. The campus is about 20 minutes ride by taxi from the airport. Participants from nearby universities could either drive or ride a bus. Both Greyhound and Barons Bus companies serve Columbus.

Please feel free to contact the organizers with any questions.
Organizers: Vitaly Bergelson (bergelson.1 at osu dot edu) and Andrey Gogolev (gogolyev.1 at osu dot edu)
The funding for the workshop is provided by Mathematics Research Institute at OSU