Distribution of distances in positive characteristic (Q2226661)

The finite version of the Erdős distinct distances problem ask for lower bounds on the cardinality \(|\mathcal{E}|\) of a point set \(\mathcal{E}\) in the vector space \(\mathbb{F}_q^d\) of dimension \(d\) over a finite field \(\mathbb{F}_q\) that guarantee that the set \(\Delta(\mathcal{E})\) of ``distances'' (defined by an expression that would give the squared distance in Euclidean spaces) equals \(\mathbb{F}_q\) or at least a substantial portion of \(\mathbb{F}_q\). For general finite fields and point sets, the fundamental result of [\textit{A. Iosevich} and \textit{M. Rudnev}, Trans. Am. Math. Soc. 359, No. 12, 6127--6142 (2007; Zbl 1145.11083)] states that \(|\mathcal{E}| \ge 4q^{(d+1)/2}\) implies \(\Delta(\mathcal{E}) = \mathbb{F}_q\). It is known that the exponent \((d+1)/2\) is optimal, at least in odd dimension, but some improvements are known when \(q = p\) is a prime number and \(\mathcal{E} = A^d\) equals the Cartesian products of a subset \(A \subset \mathbb{F}_p\). It is precisely this case that the article under review is concerned with. For dimension six or larger, bounds on \(|A^d|\) are provided that ensure \(|\Delta{A^d}| = \mathbb{F}_q\) and the number of point pairs realizing a given distance is computed up to global multiplicative constants. The proofs give rise to further results on the cardinality of the set of scalar product values and a finite field version of the Balog-Wooley Low-Energy Decomposition Theorem [\textit{A. Balog}, \textit{T. D. Wooley}, Q. J. Math. 68, No. 1, 207-226 (2017; Zbl 1435.11020)].