The stochasticity parameter of quadratic residues

TL;DR

This paper investigates the stochasticity parameter of quadratic residues modulo M, providing asymptotic analysis and showing that its normalized form fluctuates around 1 with positive density of M.

Contribution

It introduces a method to find the asymptotics of the stochasticity parameter for quadratic residues and demonstrates its oscillatory behavior relative to random sets.

Findings

The normalized stochasticity parameter of quadratic residues oscillates around 1.

The set of M where this parameter is less than 1 has positive lower density.

Asymptotic behavior of the stochasticity parameter is characterized for sets of positive density.

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) $\varliminf_{M\to\infty}\mathfrak{S}(R_M)<1<\varlimsup_{M\to\infty}\mathfrak{S}(R_M)$; b) the set $\{ M\in \mathbb{N}: \mathfrak{S}(R_M)<1 \}$ has positive lower density.