×

Sum-free subsets of finite abelian groups of type III. (English) Zbl 1358.11045

Summary: A finite abelian group \(G\) of cardinality \(n\) is said to be of type III if every prime divisor of \(n\) is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group \(G\) of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group \(G\). We supplement this result with a theorem on the structure of sum-free subsets of cardinality “close” to the largest possible in a type III abelian group \(G\).
We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of \(\operatorname{Aut}(G)\) on the set of sum-free subsets of \(G\) of the largest cardinality when \(G\) is of the form \((\mathbb{Z}/m\mathbb{Z})^r\), with all prime divisors of \(m\) congruent to 1 modulo 3, thereby extending a result of A. H. Rhemtulla and A. Penfold Street [Bull. Aust. Math. Soc. 2, 289–297 (1970; Zbl 0191.02203)]. Our second application provides an upper bound for the number of sum-free subsets of \(G\). For finite abelian groups \(G\) of type III and with a given exponent this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to B. Green and I. Z. Ruzsa [Stud. Sci. Math. Hung. 41, No. 3, 285–293 (2004; Zbl 1064.11020)].

MSC:

11B75 Other combinatorial number theory
11B30 Arithmetic combinatorics; higher degree uniformity
20K01 Finite abelian groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alon, Noga; Balogh, J.; Morris, R.; Samotij, W., Counting sum-free sets in abelian groups, Israel J. Math., 199, 1, 309-344 (2014) · Zbl 1332.11030
[2] Balasubramanian, R.; Prakash, Gyan, Asymptotic formula for sum-free sets in finite abelian groups, Acta Arith., 127, 2, 115-124 (2007) · Zbl 1127.11016
[3] Diananda, P. H.; Yap, H. P., Maximal sum-free sets of elements of finite abelian groups, Proc. Japan Acad., 45, 1-5 (1969) · Zbl 0179.04801
[4] Green, Ben, A Szemerédi-type regularity lemma in abelian groups with applications, Geom. Funct. Anal., 15, 2, 340-376 (2005) · Zbl 1160.11314
[5] Green, Ben, Counting sets with small sumset, and the clique number of random Cayley graphs, Combinatorica, 25, 3, 307-326 (2005) · Zbl 1110.11009
[6] Green, Ben; Ruzsa, Imre, Sum-free sets in abelian groups, Israel J. Math., 147, 157-189 (2005) · Zbl 1158.11311
[7] Hamidoune, Yahya ould; Plagne, Alain, A new critical pair theorem applied to sum-free sets in abelian groups, Comment. Math. Helv., 79, 183-207 (2004) · Zbl 1045.11072
[8] Lev, Vsevolod F.; Luczak, Tomasz; Schoen, Tomasz, Sum-free sets in abelian groups, Israel J. Math., 125, 347-367 (2001) · Zbl 1055.20043
[9] Prakash, Gyan, Number of sets with small sumset, and the clique number of random Cayley graph · Zbl 1373.62503
[10] Rhemtulla, A. H.; Street, A. P., Maximal sum-free sets in elementary abelian \(p\)-groups, Canad. Math. Bull., 73-80 (1971) · Zbl 0216.08903
[11] Ringel, C. M., The Birkhoff problem: How to classify subgroups of finite abelian groups?
[12] Sapozhenko, A. A., Asymptotics for the number of sum-free sets in abelian groups of even order, Dokl. Akad. Nauk, 383, 4, 454-457 (2002), (in Russian) · Zbl 1197.11131
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.