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تسجيل مستخدم جديدDiscrepancy in modular arithmetic progressions
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Celebrated theorems of Roth and of Matouv{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Theta(n^{1/4})$. 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 $Theta(n^{1/3+r_k/(6k)})$, where $r_k in {0,1,2}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.
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