Ergodic theory is concerned with the study of the dynamical behavior of measure-preserving systems and has far-reaching applications across several areas of mathematics. The purpose of this thesis is to investigate various facets of the interaction of ergodic theory with number theory and combinatorics, addressing several problems that fall within this framework. We focus on two distinct directions: combinatorial applications of ergodic theory, and multiplicative structures in dynamics.

The first topic addressed in this thesis concerns finding structured patterns in dense subsets of abelian groups. The application of ergodic theory to combinatorial problems of this nature was initiated in 1977 by Furstenbergâ s ergodic proof of Szemerédiâ s theorem. Furstenbergâ s approach has been extensively developed ever since. Building on this, Kra, Moreira, Richter and Robertson recently established a conjecture of ErdË os about infinite sumsets in dense subsets of the positive integers. In the first part of this thesis we extend their result, along with related work by Kousek and RadiÄ , from the natural numbers to abelian groups and, in some cases, to amenable groups. Our approach builds on the ergodic methods developed by Kra, Moreira, Richter, and Robertson, which we extend non-trivially to the more general setting of amenable groups.

The second topic of this thesis focuses on the connections between ergodic theory and multiplicative number theory. A striking example showcasing this connection is the well-known Sarnakâ s conjecture asserting that the Möbius (or the Liouville) function should be orthogonal to all deterministic sequences. Moreover, dynamics underlie problems in multiplicative number theory that may not initially appear dynamical at all. A prominent example is the famous conjecture of Chowla, predicting the asymptotic independence of the Liouville function at consecutive integers. Building on this perspective, Bergelson and Richter reformulated and extended several classical theorems and conjectures from number theory in the dynamical setting. Moreover, recent work by Klurman, Moreira, and Frantzikinakis has shown that the study of multiplicative measure-preserving systems has applications to problems concerning partition and density regularity of homogeneous quadratic equations. Inspired by these recent developments, the second part of this thesis is concerned with the study of multiplicative structures in dynamics. By transferring several number-theoretic concepts into the ergodic setting, we explore convergence and recurrence phenomena in multiplicative measure-preserving systems. In addition, we establish several results in multiplicative number theory, featuring theorems on the asymptotic behavior of completely multiplicative functions along linear patterns, extending a result of Klurman-Mangerel, as well as the joint distribution of Ω(n) in two consecutive integers along logarithmic averages, obtaining a natural generalization of Taoâ s theorem on the logarithmically averaged two-point Chowlaâ s conjecture.