AI 中文总结
本文证明了 Baraskar 和 Vukusic 关于迭代余数集大小 $s_j(n)/n$ 极限不存在的猜想，并引入新的迭代余数集 $T_j(n)$，证明其大小 $t_j(n)/n$ 的极限仅在 $j=0,1$ 时存在。
AI 中文摘要
对于正整数 $n$，设 $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ 并令 $s_j(n):= |S_j(n)|$。上述集合自然出现在有理数的 Pierce 级数展开长度的研究中。在文献 \cite{Ba-Vu} 中，Baraskar 和 Vukusic 猜想：对于每个固定的 $j\geq 2$，极限 $$\lim_{n\to\infty} s_j(n)/n$$ 不存在。本文肯定地证明了这一猜想。此外，我们定义了一类新的迭代余数集 $T_j(n)$，它自然产生于有理数的 Engel 级数展开长度的研究。我们类似地研究了 $t_j(n):= |T_j(n)|$ 的渐近行为。我们证明 $$\lim_{n\to\infty}\frac{t_j(n)}n$$ 恰好当 $j\in\{0,1\}$ 时存在，而对于每个固定的整数 $j\geq2$ 则不存在。
英文摘要
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 $j\geq2$.
