The Stochasticity Parameter of Quadratic Residues.

The article investigates the stochasticity parameter \( S(U) \), defined as the sum of squares of consecutive distances between elements of a subset \( U \) of the cyclic group \(\mathbb{Z}/M\mathbb{Z}\), focusing on the set \( R_M \) of quadratic residues modulo \( M \). It establishes asymptotic formulas for \( S(R_M) \) for a set \(\Omega\) of square-free integers \( M \) with controlled prime factorization, showing that the normalized stochasticity parameter \(\mathfrak{S}(R_M) = S(R_M)/s(|R_M|)\) approaches 1 as \( M \to \infty \) within \(\Omega\), where \( s(k) \) is the average stochasticity parameter over all subsets of size \( k \). The paper also demonstrates that the set of integers \( M \) for which \(\mathfrak{S}(R_M) < 1\) has positive lower density and provides explicit asymptotics for moduli of the form \( M = Ap \) with prime \( p \) and small square-free \( A \), revealing that \(\mathfrak{S}(R_M)\) can oscillate around 1 in this case. The proofs combine character sum estimates, sieve methods, and detailed analysis of gap distributions between quadratic residues, extending classical results on reduced residues and providing new insights into the additive structure of quadratic residues modulo composite moduli.