CanaDAM 2019
SFU Harbour Centre, May 29 - 31, 2019

Additive combinatorics II
Org: Hamed Hatami (McGill University)

PABLO CANDELA, Autonomous University of Madrid
A generalization of the inverse theorem for uniformity norms  [PDF]

The uniformity norms, introduced by Gowers, are very useful tools in additive combinatorics. A central result regarding these norms is the inverse theorem, proved for functions on finite cyclic groups by Green, Tao and Ziegler, which states essentially that such a function has large uniformity norm of order k+1 only if the function correlates with a nilsequence of step k. I shall discuss recent joint work with Bal√°zs Szegedy in which we obtain a generalization of the Green-Tao-Ziegler inverse theorem, extending it to a class of objects including all compact abelian groups and also more general objects such as nilmanifolds.

Non-Malleable Extractors and Codes from Additive Combinatorics  [PDF]

Extractors are algorithms that produce purely random bits from defective sources. Non-malleable extractors generalize extractors in a strong way, and produce random bits even in the presence of adversaries. I will talk about an explicit construction of non-malleable extractors using a sum-product theorem over rings. If time permits, I will discuss applications to non-malleable codes which are an elegant generalization of error-correcting codes.

This is based on joint work with David Zuckerman.

OLEKSIY KLURMAN, Royal Institute of Technology (KTH)
The Erdos discrepancy problem over the function fields  [PDF]

The famous Erdos discrepancy problem (now theorem of Tao) asserts that for any sequence $\{a_n\}_{n\ge 1}=\{\pm 1\}^{\mathbb{N}},$ \[\sup_{n,d}\left|\sum_{k=1}^na_{kd}\right|=\infty.\] It was observed during the Polymath5 project, that the analog of this statement over the polynomial ring $\mathbb{F}_q[x]$ is false. In this talk, we discuss "corrected" form of EDP over $\mathbb{F}_q[x]$ explaining some features that are not present in the number field setting. The talk is based on a joint work with A. Mangerel (CRM) and J. Teravainen (Oxford).

JOZSEF SOLYMOSI, University of British Columbia
The Uniformity Conjecture and the Sum-product Phenomenon  [PDF]

The sum-product phenomenon states that, for most of the $F(x,y)$ polynomials no matter how do we select two sets of numbers $A$ and $B$, where $|A|=|B|=n,$ the range of $F(A,B)$ will be much larger than n. We will see that assuming a major conjecture in arithmetic geometry, the Uniformity Conjecture of Bombieri and Lang, one can improve some of the classical results in this area.