Journal of the
Korean Mathematical Society JKMS

Let $\mathbb F_q$ be a finite field with $q$ elements. A function $f: \mathbb F_q^d\times \mathbb F_q^d \to \mathbb F_q$ is called a Mattila--Sj\"{o}lin type function of index $\gamma \in \mathbb R$ if $\gamma$ is the smallest real number such that whenever $|E|\geq Cq^{\gamma}$ for a sufficiently large constant $C$, the set $f(E,E):=\{f(x,y): x, y\in E\}$ is equal to $\mathbb F_q$. In this article, we construct an example of a Mattila--Sj\"{o}lin type function $f$ and provide its index, generalizing the result of Cheong, Koh, Pham and Shen [1].

Keywords: ErdH{o}s-Falconer distance problem, Falconer distance conjecture, Mattila--Sj"{o}lin type functions, finite fields

MSC numbers: Primary 52C10, 28A75, 11T23

Supported by: This work was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1I1A3049181).