On the stochasticity parameter of quadratic residues

Following V.I. Arnold, we define the stochasticity parameter $S(U)$ of a set $U\subseteq\mathbb{Z}_M$ to be the sum of squares of consecutive distances between the elements of $U$. The stochasticity parameter of the set $R_M$ of quadratic residues modulo $M$ is studied. We compare $S(R_M)$ with the average value $s(k)=s(k,M)$ of $S(U)$ over all subsets of $U\subseteq\mathbb{Z}_M$ of size $k$. It is proved that (a) for a set of moduli of positive lower density, we have $S(R_M)<s(|R_M|)$; and (b) for infinitely many moduli, $S(R_M)>s(|R_M|)$.