Numbers in a Beatty sequence which are orders only of cyclic, abelian or nilpotent groups

Abstract

Let \(C(x)\), \(A(x)\), and \(N(x)\) denote the counting functions of cyclic, abelian, and nilpotent numbers not exceeding \(x\), respectively. Their asymptotic formulas have been established in recent work by Pollack and Just. In this paper, by adapting the methods of Pollack and Just, we study the distribution of these numbers in Beatty sequences \(\mathcal{B}_{\alpha,\beta} = ([\alpha n + \beta])_{n=1}^{\infty}\), where \(\alpha > 1\) is an irrational number of finite type and \(\beta\) is a fixed real number. We prove that the counting functions \(\#C^*(x)\), \(\#A^*(x)\), and \(\#N^*(x)\) for cyclic, abelian, and nilpotent numbers in Beatty sequences satisfy asymptotic formulas that differ from those of Pollack and Just only by a factor \(1/\alpha\).