Some remarks related to the density of $\{(b^n\pmod n)/n:n\in\mathbb{N}\}$

TL;DR

This paper investigates the limit points and density of the set formed by the ratios of the remainders of powers of an integer modulo n, divided by n, providing new insights into its distribution in the interval [0,1].

Contribution

It determines the limit points of the set and establishes its density in [0,1], extending previous results from 2013.

Findings

Identifies the limit points of the set for various bases b.

Proves the set is dense in the interval [0,1].

Extends earlier work by Cilleruelo et al. on the set’s density.

Abstract

For $b\in\mathbb{N}, b\ge2$ we determine the limit points of certain subsets of $$ \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}. $$ As a consequence, we obtain the density of the latter set in $[0,1]$, a result first established in 2013 by Cilleruelo, Kumchev, Luca, Ru\'{e} and Shparlinski..