The stochasticity parameter of quadratic residues

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 which 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) ${\underline{\lim }}_{M\to \infty }\mathfrak{S}(R_M)<1<{\overline{\lim }}_{M\to \infty }\mathfrak{S}(R_M)$; b) the set $\{M\in \mathbb{N}:\mathfrak{S}(R_M)<1\}$ has positive lower density.