Structural theorems on the distance sets over finite fields

Let \(\mathbb{F}_q\) be a finite field of order \(q\). Iosevich and Rudnev (2005) proved that for any set \(A\subset \mathbb{F}_q^d\), if \(|A|\gg q^{\frac{d+1}{2}}\), then the distance set \(\Delta(A)\) contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be \(\frac{d}{2}\) in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets, the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances.