Georgia Institute of Technology College of Sciences

An open problem of interest in combinatorial number theory is that of providing a non-ergodic proof to the so called polynomial Szemeredi theorem. So far, the landmark result in this venue is that of Green who considered the emergence of 3-term arithmetic progressions whose gap is a sum of two squares (not both zero) in dense sets of integers. In view of this we consider the following problem. Given two dense subsets A and S of a finite abelian group G, what is the weakest "pseudorandomness assumption" once put on S implies that A contains a 3-term arithmetic progressions whose gap is in S? We answer this question for G=Z_n and G = F_p^n. To quantify pseudorandomness we use Gowers norms.

Graphons are limit objects that are associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible graphons. In 2011, Lovasz and Szegedy asked several questions about the complexity of the topological space of so-called typical vertices of a finitely forcible graphon can be. In particular, they conjectured that the space is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space of typical vertices is not compact. In fact, our construction actually provides an example of a finitely forcible graphon with the space which is even not locally compact. This is joint work with Roman Glebov and Dan Kral.

We study the Maker-Breaker component game, played on the edge set of a regular graph. Given a graph G, the s-component (1:b) game is defined as follows: in every round Maker claims one free edge of G and Breaker claims b free edges. Maker wins this game if her graph contains a connected component of size at least s; otherwise, Breaker wins the game. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players. To this end, we prove an extension of a theorem by Gallai-Hasse-Roy-Vitaver about graph orientations without long directed simple paths. Joint work with Alon Naor.

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties. [Joint work with Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett.]