Asymptotic Behavior of Iterated Sets of Remainders

For a positive integer $n$, let $$S_0(n)=\{1,2,\dots ,\lfloor n\mathrm{/}2\rfloor \},S_{j+1}(n)=\{nk:k\in S_j(n)\setminus \{0\}\},$$ and put $s_j(n):=\mathrm{\mid }{S}_j(n)\mathrm{\mid }$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed $j\ge 2$, the limit $$\underset{n\to \mathrm{\infty }}{\lim }{s}_j(n)\mathrm{/}n$$ does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets $T_j(n)$ that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of $t_j(n):=\mathrm{\mid }{T}_j(n)\mathrm{\mid }$. We show that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{t_j(n)}{n}$$ exists precisely for $j\in \{0,1\}$ and fails to exist for every fixed integer $j\ge 2$.