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Improved bounds for progression-free sets in \(C_8^{n}\)

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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(\(C_4^{n}\)) ≤ (3.611)n, where Cm denotes the cyclic group of order m. For finite abelian groups \(G \cong \prod\nolimits_{i = 1}^n {{C_{{m_i}}}} \), where m1,…,mn denote positive integers such that m1 |…|mn, this also yields a bound of the form \(r_3(G)\leqslant(0.903)^{{rk}_4(G)}|G|\), with rk4(G) representing the number of indices i ∈ {1,…, n} with 4 |mi. In particular, r3(\(C_8^{n}\)) ≤ (7.222)n. In this paper, we provide an exponential improvement for this bound, namely r3(\(C_8^{n}\)) ≤ (7.0899)n.

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Correspondence to Cosmin Pohoata.

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Research supported by Russian Science Foundation grant 17-71-20153.

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Petrov, F., Pohoata, C. Improved bounds for progression-free sets in \(C_8^{n}\). Isr. J. Math. (2020). https://doi.org/10.1007/s11856-020-1977-0

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