论文标题
二次残基的随机性参数
The stochasticity parameter of quadratic residues
论文作者
论文摘要
在V. I. Arnold之后，我们定义了$ \ Mathbb {Z}/M \ Mathbb {Z} $的子集$ u $的随机性参数$ s（u）$，是$ u $的元素之间连续距离的正方形的总和。在本文中，我们研究了集合$ r_m的二次残基$ m $ m $的随机性参数。我们提出了一种方法，该方法允许找到一组正密度的$ S（R_M）$的渐近学。特别是，我们获得以下两个推论。用$ s（k）= s（k，\ mathbb {z}/m \ mathbb {z}）$ $ s（u \ subseteq \ subseteq \ subseteq \ subseteq \ subseteq \ subseteq \ mathbb {z}/m \ mathbb {z} $ k $ a $ k $的平均值$ s（u）$，$ s $ k $ a的$ k $ a $ k $ k $ k $ k $ k $ k $。令$ \ mathfrak {s}（r_m）= s（r_m）/s（| r_m |）$。我们表明a）$ \ varliminf_ {m \ to \ infty} \ mathfrak {s}（r_m）<1 <\ varlimsup_ {m \ to \ infty} \ infty} \ mathfrak {s s}（s}（r_m）$; b）集合$ \ {m \ in \ mathbb {n}：\ mathfrak {s}（r_m）<1 \} $具有较低的较低密度。
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.
