Doowon Koh (고두원), Mattila-Sjölin type functions: A finite field model

Let ${\mathbb{F}}_q$ be a finite field of order $q$ which is a prime power. In the finite field setting, we say that a function $\varphi :{\mathbb{F}}_q^d\times {\mathbb{F}}_q^d\to {\mathbb{F}}_q$ is a Mattila-Sjölin type function in ${\mathbb{F}}_q^d$ if for any $E\subset {\mathbb{F}}_q^d$ with $|E|\gg q^{\frac{d}{2}}$, we have $\varphi (E,E)={\mathbb{F}}_q$. The main purpose of this talk is to present the existence of such a function. More precisely, we will construct a concrete function $\varphi :{\mathbb{F}}_q^4\times {\mathbb{F}}_q^4\to {\mathbb{F}}_q$ with $q\equiv 3\mathrm{mod}4$ such that if $E\subset {\mathbb{F}}_q^4$ with $|E|>q^2,$ then $\varphi (E,E)={\mathbb{F}}_q$. This is a joint work with Daewoong Cheong, Thang Pham, and Chun-Yen Shen.