On the number of subsequence sums related to the support of a sequence in finite abelian groups

2024/04/17 by Rui Wang, Wang, Rui, Han Chao +3 · 1 citation
Computer Science · Engineering · Mathematics · #11B13 #11B50 #11P70 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2404.11307

openalex publication_date 2024/04/17 · openalex created_date 2024/04/19 · openalex updated_date 2026/07/28

Abstract

Let G be a finite abelian group and S a sequence with elements of G. Let |S| denote the length of S and supp(S) the set of all the distinct terms in S. For an integer k with k∈ [1, |S|], let Σ_k(S) ⊂ G denote the set of group elements which can be expressed as a sum of a subsequence of S with length k. Let Σ(S)=∪_k=1^|S|Σ_k(S) and Σ_{≥ k}(S)=∪_t=k^|S|Σ_t(S). It is known that if 0\not∈ Σ(S), then |Σ(S)|≥ |S|+|supp(S)|-1. In this paper, we determine the structure of a sequence S satisfying 0∉ Σ(S) and |Σ(S)|= |S|+|supp(S)|-1. As a consequence, we can give a counterexample of a conjecture of Gao, Grynkiewicz, and Xia. Moreover, we prove that if |S|>k and 0\not∈ Σ_{≥ k}(S)∪ supp(S), then |Σ_{≥ k}(S)|≥ |S|-k+|supp(S)|. Then we can give an alternative proof of a conjecture of Hamidoune, which was first proved by Gao, Grynkiewicz, and Xia.

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