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

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..