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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.
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Sets without k ‐term progressions can have many shorter progressions
Abstract Letfs, k(n)be the maximum possible number ofs‐term arithmetic progressions in a set ofnintegers which contains nok‐term arithmetic progression. For all fixed integersk > s ≥ 3, we prove thatfs, k(n) = n2 − o(1), which answers an old question of Erdős. In fact, we prove upper and lower bounds forfs, k(n)which show that its growth is closely related to the bounds in Szemerédi's theorem.
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- Award ID(s):
- 1855635
- PAR ID:
- 10240192
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 58
- Issue:
- 3
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 383-389
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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