Discrepancy in modular arithmetic progressions. Issue 11 (1st November 2022)

• Record Type: Journal Article
• Authors: Fox, Jacob; Xu, Max Wenqiang; Zhou, Yunkun
• Is Part Of: Compositio mathematica. Volume 158:Issue 11(2022)
• Page Start: 2082
• Page End: 2108
• Publication Date: 2022-11-01
• Subjects: discrepancy -- congruence -- arithmetic progression -- partial coloring method -- Fourier analysis; 11K38 -- 11B25 -- 11A07 -- 11B30 -- 05D40 -- 05D10 -- 11A25 -- 11B50; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X22007758
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $\Theta (n^{1/4})$ . 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.