Discrepancy in modular arithmetic progressions

TL;DR

This paper investigates the discrepancy of arithmetic progressions in the cyclic group ting the asymptotic behavior for all positive integers and provides precise bounds for prime power moduli, extending classical results.

Contribution

It determines the asymptotic logarithm of the discrepancy in ting all positive integers and precisely characterizes the discrepancy for prime power moduli, solving an open problem.

Findings

Discrepancy in ting is pproximate to n^{1/3 + r_k/(6k)} for prime powers

Logarithm of discrepancy is asymptotically determined for all n

Solved a problem posed by Hebbinghaus and Srivastav

Abstract

Celebrated theorems of Roth and of Matou\v{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.