arXiv subjects
Publications and source records attributed to A. Lemos.
Let $G$ be a group and $A\subseteq [1,\exp(G)-1]$. We define the constant ${\sf C}_A(G),$ which is the least positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has an $A$-weighted consecutive product-one subsequence. In this paper, among other things, we prove that ${\sf C}_A(C_n^2)=4$ with $A=[1,n-1],$ and ${\sf C}(H\times K)=|H||K|$, where $H$ is a finite abelian group and $K$ is a metacyclic group.
Let $G$ be a finite additive abelian group with exponent $d^kn, d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\ldots,d^kn-1\}\setminus\{d^kn/d^i:i\in[1,k]\}, $ we investigate the lower bound of the number $N_{A,0}(S)$, which denotes the number of $A$-weighted zero-sum subsequences of $S.$ In particular, we prove that $N_{A,0}(S)\ge 2^{|S|-D_A(G)+1},$ where $D_A(G)$ is the $A$-weighted Davenport Constant. We also characterize the structures of the extremal sequences for which equality holds for some groups.
Let $G$ be a finite additive abelian group. For given $k$ a positive integer, the $k$-Harborth constant $g^k(G)$ is defined to be the smallest positive integer $t$ such that given a set $S$ of elements of $G$ with size $t$ there exists a zero-sum subset of size $k$. We find either the exact value of $g^k(G)$, or lower and upper bounds for this constant for some groups.