2019/03/08 by Тарас Банах, Banakh, Taras, Alex Ravsky +1
Engineering · Mathematics · #05E15 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1903.03577
openalex publication_date 2019/03/08 · openalex created_date 2019/03/22 · openalex updated_date 2026/07/28
A subset D of an Abelian group is decomposable if ∅≠ D⊂ D+D. In the paper we give partial answer to an open problem asking whether every finite decomposable subset D of an Abelian group contains a non-empty subset Z⊂ D with ∑ Z=0. For every n∈\mathbb N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n-1 such that ∑ D=0, but ∑ T≠ 0 for any proper non-empty subset T⊂ D. On the other hand, we prove that every decomposable subset D⊂\mathbb R of cardinality |D|≤ 7 contains a non-empty subset Z⊂ D of cardinality |Z|≤\frac12|D| with ∑ Z=0. For every n∈\mathbb N we present a subset D⊂\mathbb Z of cardinality |D|=2n such that ∑ Z=0 for some subset Z⊂ D of cardinality |Z|=n and ∑ T≠ 0 for any non-empty subset T⊂ D of cardinality |T|