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Title: Discrepancy in modular arithmetic progressions
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.  more » « less
Award ID(s):
2154129 1855635
NSF-PAR ID:
10431297
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Compositio Mathematica
Volume:
158
Issue:
11
ISSN:
0010-437X
Page Range / eLocation ID:
2082 to 2108
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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