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arXiv · 2204.07345

Extremal sequences for the unit-weighted Gao constant of $\mathbb Z_n$

Santanu Mondal·Krishnendu Paul·Shameek Paul
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Abstract

For $A\subseteq \mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero. Sequences of length $E_A(n)-1$ in $\mathbb Z_n$, which do not have any $A$-weighted zero-sum subsequence of length $n$ are called $A$-extremal sequences for the Gao constant. Such a sequence which has $n-1$ zeroes is said to be of the standard type. When $A=U(n)$ (units in $\mathbb Z_n$) where $n$ is odd, we characterize all such sequences and show that they are of the standard type. When $n$ is even, we give examples of such sequences which are not of the standard type. We also characterize the $U(n)$-extremal sequences for the Gao constant, when $n=2^rp$, where $p$ is an odd prime.

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Santanu Mondal, Krishnendu Paul, Shameek Paul. 2022-04-15. Extremal sequences for the unit-weighted Gao constant of $\mathbb Z_n$. https://arxiv.org/abs/2204.07345

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