The Stochasticity Parameter of Quadratic Residues

Abstract

Abstract Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a subset $U$ of ${\mathbb {Z}}/M{\mathbb {Z}}$ to be the sum of squares of the consecutive distances between elements of $U$. In this paper, we study the stochasticity parameter of the set $R_{M}$ of quadratic residues modulo $M$. We present a method that allows to find the asymptotics of $S(R_{M})$ for a set of $M$ of positive density. In particular, we obtain the following two corollaries. Denote by $s(k)=s(k,{\mathbb {Z}}/M{\mathbb {Z}})$ the average value of $S(U)$ over all subsets $U\subseteq {\mathbb {Z}}/M{\mathbb {Z}}$ of size $k$, which can be thought of as the stochasticity parameter of a random set of size $k$. Let ${\mathfrak {S}}(R_{M})=S(R_{M})/s(|R_{M}|)$. We show that (a) $\varliminf _{M\to \infty }{\mathfrak {S}}(R_{M})<1<\varlimsup _{M\to \infty }{\mathfrak {S}}(R_{M})$; (b) the set $\{ M\in {\mathbb {N}}: {\mathfrak {S}}(R_{M})<1 \}$ has positive lower density.