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

For $b\in \mathbb{N},b\ge 2$ we determine the limit points of certain subsets of $$\{\frac{b^n(\mathrm{m}\mathrm{o}\mathrm{d}n)}{n}:n\in \mathbb{N}\}\mathrm{.}$$ As a consequence, we obtain the density of the latter set in $[0,1]$, a result first established in 2013 by Cilleruelo, Kumchev, Luca, Rué and Shparlinski…