标题： 余数集合的界 显示英文标题

标题： Bounds for sets of remainders

摘要： 设$s(n)$为不同的余数$n \bmod k$的个数，其中$1 \leq k \leq \lfloor n/2 \rfloor$。 这个相当自然的序列是OEIS中的序列A283190，虽然已知一些基本事实，但似乎它几乎没有被研究过。 首先，我们证明$s(n) = c \cdot n + O(n/(\log n \log \log n))$，其中$c$是一个显式的常数。 然后我们关注相邻项$s(n)$和$s(n+1)$之间的差异。 结果表明，该值最多只能增加1，但存在任意大的减少。 我们证明差异被$O(\log \log n)$限制。 最后，我们考虑“迭代余数集”。 这些与来自Pierce展开的问题有关，我们同样证明了这些集合的大小的界限。 显示英文摘要

摘要： Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that $s(n) = c \cdot n + O(n/(\log n \log \log n))$, where $c$ is an explicit constant. Then we focus on differences between consecutive terms $s(n)$ and $s(n+1)$. It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the differences are bounded by $O(\log \log n)$. Finally, we consider ''iterated remainder sets''. These are related to a problem arising from Pierce expansions, and we prove bounds for the size of these sets as well.