A note on the number of distinct elements and zero-sum subsequence lengths in cyclic groups

2025/05/07 by Pop, Claudiu, Ţurcaş, George C.
#11B75 #11R27 #20K01 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.04187

Abstract

In this short note we investigate zero-sum sequences in finite abelian groups, examining the relationship between the sequence's support size, that is the number of distinct elements, and its properties concerning zero-sums. In particular, for sequences S in a cyclic group, we establish a direct connection between MZ(S), the length of the shortest nonempty subsequence summing to zero and the number of distinct values in S. Our results reveal that sequences with larger support must contain shorter non-empty zero-sum subsequences, in line with classical zero-sum results. Additionally, we present one application of our main result to a factorization of ideals problem in rings of integers of a number field.

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