Discrepancy in modular arithmetic progressions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917629044719616 |
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| author | Fox, Jacob Xu, Max Wenqiang Zhou, Yunkun |
| author_facet | Fox, Jacob Xu, Max Wenqiang Zhou, Yunkun |
| contents | Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Θ(n^{1/4})$. We study the analogous problem in the $\mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ is $Θ(n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_03929 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Discrepancy in modular arithmetic progressions Fox, Jacob Xu, Max Wenqiang Zhou, Yunkun Combinatorics Number Theory Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Θ(n^{1/4})$. We study the analogous problem in the $\mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ is $Θ(n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav. |
| title | Discrepancy in modular arithmetic progressions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2104.03929 |