Weak Freiman isomorphisms and sequencings of small sets

2024/07/22 by Simone de Melo Costa, Costa, Simone, Stefano Della Fiore +1
Biochemistry, Genetics and Molecular Biology · Computer Science · #11B75 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Fibroblast Growth Factor Research #Genetic Syndromes and Imprinting

paper · pdf · doi:10.48550/arxiv.2407.15785

openalex publication_date 2024/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce a weakening of the Freiman isomorphisms between subsets of non necessarily abelian groups. Inspired by the breakthrough result of Kravitz, [14], on cyclic groups, as a first application, we prove that any subset of size k of the dihedral group D_2m (and, more in general, of a class of semidirect products) is sequenceable, provided that the prime factors of m are larger than k!. Also, a refined bound of k!/2 for the size of the prime factors of m can be obtained for cyclic groups ℤ_m, slightly improving the result of [14]. Then, applying again the concept of weak Freiman isomorphism, we show that any subset of size k of the dicyclic group Dic_m is sequenceable, provided that the prime factors of m are larger than k^k.

Related

• Freiman's Theorem in an arbitrary abelian group
• Sets with small sumset and rectification
• On a Theorem of Deshouillers and Freiman
• Some remarks on products of sets in the Heisenberg group and in the affine group
• Sequencings in Semidirect Products via the Polynomial Method