Discrepancy in modular arithmetic progressions

Abstract

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.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1: Section 4.6 in Mat
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Theorem 3.1
• Lemma 3.2