Bounds for sets of remainders
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916923676033024 |
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| author | Baraskar, Omkar Vukusic, Ingrid |
| author_facet | Baraskar, Omkar Vukusic, Ingrid |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20853 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds for sets of remainders Baraskar, Omkar Vukusic, Ingrid Number Theory 11N37, 11A07, 11B83 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. |
| title | Bounds for sets of remainders |
| topic | Number Theory 11N37, 11A07, 11B83 |
| url | https://arxiv.org/abs/2508.20853 |