Geodesic
On the stochasticity parameter of quadratic residues
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 491 (2020), pp. 19-22

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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 $S(R_M)$; and (b) for infinitely many moduli, $S(R_M)>s(|R_M|)$.
Keywords: quadratic residues, stochasticity parameter. 
@article{DANMA_2020_491_a3,
     author = {M. R. Gabdullin},
     title = {On the stochasticity parameter of quadratic residues},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {19--22},
     publisher = {mathdoc},
     volume = {491},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_491_a3/}
}
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M. R. Gabdullin. On the stochasticity parameter of quadratic residues. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 491 (2020), pp. 19-22. http://geodesic.mathdoc.fr/item/DANMA_2020_491_a3/