Estimation of the sum of powers of distances between residues of prime modulus
Let \(A\) be a set of elements in the ring \(\mathbb{Z}_{n}\), \(A=\{a_{i}\}^{t}_{i=1}\), \(0\leq a_{1}<\ldots<a_{t}<n\), and \(a_{t+1}:=a_{1}+n\). V. I. Arnold introduced the so called parameter of stochasticity \(R(A)=\displaystyle\sum_{j=1}^{t} (a_{j+1}-a_{j})^{2}\). Let \(q >0\). The author studies the generalized parameter \(R_{q}(A)=\displaystyle\sum_{j=1}^{t} (a_{j+1}-a_{j})^{q}\), where \(n=p\) is a prime number, and \(A\) is a subgroup in \(\mathbb{Z}_{p}\setminus \{0\}\). He proves, that if \(p\rightarrow\infty\), \(t>p^{1/2}\), and \(q\geq \frac{105}{64}\), then \(R_{q}(g)\ll p^{\frac{463}{504}q+o(1)}\).
scientific article; zbMATH DE number 6229096