Long paths in the distance graph over large subsets of vector spaces over finite fields (Q2794875)

Let \(E\) denote a subset of \(\mathbb{F}_q^d\), the \(d\)-dimensional column vector space over the \(q\)-element field \(\mathbb{F}_q\). For \(x,y\in E\), their distance \(\|x-y\|\) is defined as the sum of squares of differences in their coordinates, \(\sum_{i=1}^d (x_i-y_i)^2\). The paper counts asymptotically the sequences \(P_1,P_2,\ldots,P_{k+1}\) in \(E\) such that \(\|P_{i+1}-P_i\|=t_i\), for a prescribed sequence \(t_1,\ldots,t_k\) from \(\mathbb{F}_q\). The main term of the asymptotics is \(|E|^{k+1}/q^k\), with a much smaller error term, if \(|E|\) is sufficiently big. In other words, the distribution of sequences regarding distances is near uniform. For a corollorary, if \(|E|>{2k\over\ln 2}q^{d+1\over 2}\), such a \(P_1,P_2,\ldots,P_{k+1}\) sequence in \(E\) always exist.