Search

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. [ABSTRACT FROM AUTHOR]