×

zbMATH — the first resource for mathematics

Progression-free sets in \(\mathbb{Z}_4^n\) are exponentially small. (English) Zbl 1425.11019
Summary: We show that for an integer \(n\geq 1\), any subset \(A\subseteq\mathbb{Z}_4^n\) free of three-term arithmetic progressions has size \(|A|\leq 4^{\gamma n}\), with an absolute constant \(\gamma\approx 0.926\).

MSC:
11B30 Arithmetic combinatorics; higher degree uniformity
11B25 Arithmetic progressions
51E20 Combinatorial structures in finite projective spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Bateman and N. H. Katz, ”New bounds on cap sets,” J. Amer. Math. Soc., vol. 25, iss. 2, pp. 585-613, 2012. · Zbl 1262.11010
[2] T. F. Bloom, ”A quantitative improvement for Roth’s theorem on arithmetic progressions,” J. Lond. Math. Soc., vol. 93, iss. 3, pp. 643-663, 2016. · Zbl 1364.11024
[3] J. Bourgain, ”On triples in arithmetic progression,” Geom. Funct. Anal., vol. 9, iss. 5, pp. 968-984, 1999. · Zbl 0959.11004
[4] T. C. Brown and J. P. Buhler, ”A density version of a geometric Ramsey theorem,” J. Combin. Theory Ser. A, vol. 32, iss. 1, pp. 20-34, 1982. · Zbl 0476.51008
[5] P. Frankl, R. L. Graham, and V. Rödl, ”On subsets of abelian groups with no \(3\)-term arithmetic progression,” J. Combin. Theory Ser. A, vol. 45, iss. 1, pp. 157-161, 1987. · Zbl 0613.10043
[6] D. R. Heath-Brown, ”Integer sets containing no arithmetic progressions,” J. London Math. Soc., vol. 35, iss. 3, pp. 385-394, 1987. · Zbl 0589.10062
[7] V. F. Lev, ”Progression-free sets in finite abelian groups,” J. Number Theory, vol. 104, iss. 1, pp. 162-169, 2004. · Zbl 1043.11022
[8] V. F. Lev, ”Character-free approach to progression-free sets,” Finite Fields Appl., vol. 18, iss. 2, pp. 378-383, 2012. · Zbl 1284.11020
[9] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North-Holland Publ. Co., 1977. · Zbl 0369.94008
[10] R. Meshulam, ”On subsets of finite abelian groups with no \(3\)-term arithmetic progressions,” J. Combin. Theory Ser. A, vol. 71, iss. 1, pp. 168-172, 1995. · Zbl 0832.11006
[11] K. Roth, ”Sur quelques ensembles d’entiers,” C. R. Acad. Sci. Paris, vol. 234, pp. 388-390, 1952. · Zbl 0046.04302
[12] K. Roth, ”On certain sets of integers,” J. London Math. Soc., vol. 28, pp. 104-109, 1953. · Zbl 0050.04002
[13] T. Sanders, ”Roth’s theorem in \(\mathbb Z^n_4\),” Anal. PDE, vol. 2, iss. 2, pp. 211-234, 2009. · Zbl 1197.11017
[14] T. Sanders, ”On certain other sets of integers,” J. Anal. Math., vol. 116, pp. 53-82, 2012. · Zbl 1280.11009
[15] T. Sanders, ”On Roth’s theorem on progressions,” Ann. of Math., vol. 174, iss. 1, pp. 619-636, 2011. · Zbl 1264.11004
[16] E. Szemerédi, ”Integer sets containing no arithmetic progressions,” Acta Math. Hungar., vol. 56, iss. 1-2, pp. 155-158, 1990. · Zbl 0721.11007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.