On unit-weighted zero-sum constants of $\mathbb Z_n$
Santanu Mondal, Krishnendu Paul, Shameek Paul

TL;DR
This paper investigates the minimal sequence length in cyclic groups ensuring the existence of weighted zero-sum subsequences with consecutive terms, providing new proofs and characterizations for specific cases including powers of two and particular subsets.
Contribution
It offers a new proof for the value of $C_{U(n)}(n)$ for all $n$, characterizes extremal sequences for powers of two, and determines constants for specific subsets of $Z_n$.
Findings
Determined $C_{U(n)}(n)$ for all $n$ using a new argument.
Characterized $C$-extremal sequences when $n$ is a power of 2.
Calculated $C_A(n)$ for sets of all odd/even elements and initial segments of integers.
Abstract
Given , the constant is defined to be the smallest natural number such that any sequence of elements in has an -weighted zero-sum subsequence having consecutive terms. The value of is known when is odd. We give a different argument to determine the value of for any . A -extremal sequence for is a sequence in whose length is and which does not have any -weighted zero-sum subsequence having consecutive terms. We characterize the -extremal sequences for when is a power of 2. For any , we determine the value of where is the set of all odd (or all even) elements of and also when where .
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Taxonomy
TopicsMathematical Approximation and Integration · Advanced Banach Space Theory · Advanced Harmonic Analysis Research
