Huy Tuan Pham

Additive combinatorics from a probabilistic and combinatorial perspective

Abstract: Sumset is a fundamental object in additive combinatorics. Since Freiman’s celebrated theorem which characterizes the structure of integer sets with small sumsets, the structure theory of sets with small sumsets has gathered significant development, breakthroughs and applications.

I will discuss new combinatorial and probabilistic approaches to the study of sets with small doubling. Motivated from the study of random Cayley graphs, these perspectives provide new ways to quantify the structure and complexity of sets with small sumsets that are robust, flexible and quantitative. Utilizing the rich interaction between additive combinatorics and probabilistic combinatorics, these approaches lead to new understanding of random Cayley graphs, as well as the resolution of longstanding conjectures in additive combinatorics, including Ruzsa’s conjecture on sets with small doubling in abelian groups with bounded exponent. We will also discuss applications to the enumeration of sets with small doubling, and an emerging direction to study random graph models with significant symmetries and dependencies.