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 the consecutive distances between elements of $U$. We study the stochasticity parameter of the set $R_M$ of quadratic residues modulo $M$. Denote by $s(k)=s(k,\mathbb{Z}_M)$ the average value of $S(U)$ over all subsets $U\subseteq \mathbb{Z}_M$ of size $k$, which can be thought of as the stochasticity parameter of a random set of size $k$. We prove that
a) $\varliminf_{M\to\infty}\frac{S(R_M)}{s(|R_M|)}<1<\varlimsup_{M\to\infty}\frac{S(R_M)}{s(|R_M|)}$;
b) the set $\{ M\in \mathbb{N}: S(R_M)<s(|R_M|) \}$ has positive lower density.

^* Zoom Meeting ID: 979 5396 3790