Asymptotic Behavior of Iterated Sets of Remainders

Abstract

For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. 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 $j\geq 2$, the limit $$\lim_{n\to\infty} s_j(n)/n$$ 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 $t_j(n) := |T_j(n)|$. We show that $$\lim_{n\to\infty}\frac{t_j(n)}n$$ exists precisely for $j\in\{0,1\}$ and fails to exist for every fixed integer $j\geq2$.