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Title:ÌýConnections between ergodic theory and multiplicative numberÌýtheory
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Abstract:ÌýIn this talk, we will discuss several connections between ergodic theory and multiplicative number theory. Given a probability space (X, A, µ), a multiplicative action S = (Sn)n∈N on (X, A, µ) is a sequence of measure-preserving transformations Sn : X → X satisfying Snm = Sn â—¦ Sm for all n, m ∈ N. This object can be viewed as the natural ergodic analog of a completely multiplicative function f : N → S 1 . We will present new results on multiplicative actions, applications of ergodic methods to problems in number theory, and open questions in both areas. In particular, we will focus on recurrence phenomena for multiplicative actions and their numbertheoretic consequences. Namely, we characterize the integers a, b, c, d for whichÌýlim inf n→∞ |f(an + b) − f(cn + d)| = 0Ìýholds for all completely multiplicative functions f : N → S 1 . This extends a theorem of Klurman and Mangerel corresponding to the pair (n, n + 1). The talk will conclude with a discussion of how ergodic methods can be employed to address problems on the partition regularity of homogeneous quadratic equations, presenting current progress and highlighting the challenges involved.

Venue: Concordia University LB 921-04