Uncommon and Sidorenko systems of equations

Wednesday, 28 June 2023

Abstract

A system of linear equations $L$ over the finite field $\mathbb{F}_q$ is {\em common} if the number of solutions to $L$ in any two-colouring of $\mathbb{F}_q^n$  is asymptotically (as $n\to\infty$) at least the expected number of monochromatic solutions in a random colouring of $\mathbb{F}_q^n.$ For example, a Schur triple $x+y=z$ was shown to be common by Cameron, Cilleruelo and Serra in 2007. Another heavily studied specific example is that of an arithmetic progression of length four (4-AP), which can be described by two equations of the form $x - 2y + z = 0$ and $y - 2z + w = 0.$ Wolf showed that these are uncommon (over $Z_N$ and over $\mathbb{F}_5$). 

Motivated by these existing results on specific systems, as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building on earlier work of Cameron, Cilleruelo and Serra, common linear equations have been fully characterised by Fox, Pham and Zhao

In this talk, we discuss recent progress towards characterising common systems of two or more equations. In particular, we prove that any system containing a 4-AP is uncommon, confirming a conjecture of Saad and Wolf. We also discuss the related concept of Sidorenko systems of equations.

Joint work with Nina Kam\v{c}ev and Natasha Morrison.