Products of Differences over Arbitrary Finite Fields

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then $$|(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| > \frac{q}{2}.$$ Any $\delta < 1/13,542$ suffices for sufficiently large $q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. Our proof is based on a qualitatively optimal characterisation of sets $A,X \subseteq \mathbb{F}_q$ for which the number of solutions to the equation $$(a_1-a_2) = x (a_3-a_4) \, , \; a_1,a_2, a_3, a_4 \in A, x \in X$$ is nearly maximum. A key ingredient is determining exact algebraic structure of sets $A, X$ for which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and Glibichuk using work of Gill, Helfgott, and Tao. We also prove a stronger statement for $$(A-B)(C-D) = \{ (a -b) (c-d) : a \in A, b \in B, c \in C, d \in D\}$$ when $A,B,C,D$ are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.Comment: 42 page

Interactions between Ergodic Theory and Combinatorial Number Theory

The seminal work of Furstenberg on his ergodic proof of Szemerédi’s Theorem gave rise to a very rich connection between Ergodic Theory and Combinatorial Number Theory (Additive Combinatorics). The former is concerned with dynamics on probability spaces, while the latter is concerned with Ramsey theoretic questions about the integers, as well as other groups. This thesis further develops this symbiosis by establishing various combinatorial results via ergodic techniques, and vice versa. Let us now briefly list some examples of such. A shorter ergodic proof of the following theorem of Magyar is given: If B Zd, where d 5, has upper Banach density at least > 0, then the set of all squared distances in B, i.e., the set fkb1 b2k2 j b1; b2 2 Bg, contains qZ>R for some integer q = q( ) > 0 and R = R(B). Our technique also gives rise to results on the abundance of many other higher order Euclidean configurations in such sets. Next, we turn to establishing analogues of this result of Magyar, where k k2 is replaced with other quadratic forms and various other algebraic functions. Such results were initially obtained by Björklund and Fish, but their techniques involved some deep measure rigidity results of Benoist-Quint. We are able to recover many of their results and prove some completely new ones (not obtainable by their techniques) in a much more self-contained way by avoiding these deep results of Benoist-Quint and using only classical tools from Ergodic Theory. Finally, we extend some recent ergodic analogues of the classical Plünnecke inequalities for sumsets obtained by Björklund-Fish and establish some estimates of the Banach density of product sets in amenable non-abelain groups. We have aimed to make this thesis accesible to readers outside of Ergodic Theory who may be primarily interested in the arithmetic and combinatorial applications

Classical and modern approaches for Plünnecke-type inequalities

The main objective of this thesis is to present and prove Plünnecke's Inequality, a theorem that gives bounds for sumsets in commutative groups. An introduction to the theory of set addition is presented. Three different proofs of Plünnecke's Inequality are presented, two of them relying strongly on graph theory, the third being more elementary. Some other tools and techniques are introduced to obtain generalizations of Plünnecke's Inequality. The most important are the Plünnecke-Ruzsa Inequality, that gives bounds to general sum-and-difference sets in commutative groups, and some generalizations to the non-commutative case, related to Tao's Theorem. Some other generalizations involve the sum of different sets. The results are used to prove an important structural result