TY - JOUR

T1 - Improved bounds for progression-free sets in C8n

AU - Petrov, Fedor

AU - Pohoata, Cosmin

N1 - Funding Information: Research supported by Russian Science Foundation grant 17-71-20153. Publisher Copyright: © 2020, The Hebrew University of Jerusalem. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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(C4n) ≤ (3.611)n, where Cm denotes the cyclic group of order m. For finite abelian groups G≅∏i=1nCmi, where m1,…,mn denote positive integers such that m1 |…|mn, 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 |mi. In particular, r3(C8n) ≤ (7.222)n. In this paper, we provide an exponential improvement for this bound, namely r3(C8n) ≤ (7.0899)n.

AB - 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(C4n) ≤ (3.611)n, where Cm denotes the cyclic group of order m. For finite abelian groups G≅∏i=1nCmi, where m1,…,mn denote positive integers such that m1 |…|mn, 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 |mi. In particular, r3(C8n) ≤ (7.222)n. In this paper, we provide an exponential improvement for this bound, namely r3(C8n) ≤ (7.0899)n.

UR - http://www.scopus.com/inward/record.url?scp=85079500351&partnerID=8YFLogxK

U2 - 10.1007/s11856-020-1977-0

DO - 10.1007/s11856-020-1977-0

M3 - Article

AN - SCOPUS:85079500351

VL - 236

SP - 345

EP - 363

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -