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Improved Bounds for Progression-Free Sets in C^n₈

Petrov, Fedor and Pohoata, Cosmin (2020) Improved Bounds for Progression-Free Sets in C^n₈. Israel Journal of Mathematics, 236 (2). pp. 345-363. ISSN 0021-2172.

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Let G be a finite group, and let r₃(G) represent the size of the largest subset of G without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that r₃(C₄^n) ≤ (3.611)^n, where C_m denotes the cyclic group of order m. For finite abelian groups G≅∏^n_(i=1), where m₁,…,m_n denote positive integers such that m₁ |…|m_n, this also yields a bound of the form r₃(G)⩽(0.903)^(rk₄(G))|G|, with rk₄(G) representing the number of indices i ∈ {1,…, n} with 4 |m_i. In particular, r₃(Cn₈) ≤ (7.222)^n. In this paper, we provide an exponential improvement for this bound, namely r₃(Cn₈) ≤ (7.0899)^n.

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Additional Information:© 2020 The Hebrew University of Jerusalem. First Online: 12 February 2020. Research supported by Russian Science Foundation grant 17-71-20153.
Funding AgencyGrant Number
Russian Science Foundation17-71-20153
Issue or Number:2
Record Number:CaltechAUTHORS:20200110-155400957
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Official Citation:Petrov, F., Pohoata, C. Improved bounds for progression-free sets in Cn8. Isr. J. Math. 236, 345–363 (2020).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100648
Deposited By: Tony Diaz
Deposited On:11 Jan 2020 00:30
Last Modified:05 May 2020 20:48

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