On unit weighted zero-sum constants of $\mathbb Z_n$

Santanu Mondal, Krishnendu Paul, Shameek Paul

Published 2022-03-05Version 1

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. When $A=U(n)$ (the set of all units in $\mathbb Z_n$), the value of $E_A(n)$ has been determined by Simon Griffiths and by Florian Luca. We give another proof of this result and also determine the values of two related constants $C_A(n)$ and $D_A(n)$, by using the corresponding constants $E(\mathbb Z_2^a),C(\mathbb Z_2^a)$ and $D(\mathbb Z_2^a)$ for the group $\mathbb Z_2^a$. We also characterize all sequences of length $E_A(n)-1$ in $\mathbb Z_n$, which do not have any $A$-weighted zero-sum subsequence of length $n$, when $n$ is a power of 2 and $A=U(n)$.