Published May 2021
| Submitted
Journal Article
Open
Sets without k-term progressions can have many shorter progressions
- Creators
- Fox, Jacob
- Pohoata, Cosmin
Abstract
Let f_(s, k)(n) be the maximum possible number of s‐term arithmetic progressions in a set of n integers which contains no k‐term arithmetic progression. For all fixed integers k > s ≥ 3, we prove that f_(s, k)(n) = n^(2 − o(1)), which answers an old question of Erdős. In fact, we prove upper and lower bounds for f_(s, k)(n) which show that its growth is closely related to the bounds in Szemerédi's theorem.
Additional Information
© 2020 Wiley Periodicals LLC. Issue Online: 10 March 2021; Version of Record online: 15 December 2020; Manuscript accepted: 27 July 2020; Manuscript received: 10 April 2020. Funding Information: NSF Grant Number: DMS‐1855635.Attached Files
Submitted - 1908.09905.pdf
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Additional details
- Eprint ID
- 100642
- Resolver ID
- CaltechAUTHORS:20200110-150751274
- DMS‐1855635
- NSF
- Created
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2020-01-11Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field