23 research outputs found
Davenport constant with weights
For the cyclic group and any non-empty
. We define the Davenport constant of with weight ,
denoted by , to be the least natural number such that for any
sequence with , there exists a non-empty
subsequence and such that
. Similarly, we define the constant to be
the least such that for all sequences with
, there exist indices , and with . In the present paper, we show that
. This solve the problem raised by Adhikari and Rath
\cite{ar06}, Adhikari and Chen \cite{ac08}, Thangadurai \cite{th07} and
Griffiths \cite{gr08}.Comment: 6page
Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory
We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group , a set of weights , and an integral parameter , the -wise Davenport constant with weights is the smallest integer such that each sequence over of length has at least disjoint zero-subsums with weights . And, for an integral parameter , the -constrained Davenport constant with weights is the smallest such that each sequence over of length has a zero-subsum with weights of size at most . First, we establish a link between these two types of constants and several basic and general results on them. Then, for elementary -groups, establishing a link between our constants and the parameters of linear codes as well as the cardinality of cap sets in certain projective spaces, we obtain various explicit results on the values of these constants
Arithmetic-Progression-Weighted Subsequence Sums
Let be an abelian group, let be a sequence of terms
not all contained in a coset of a proper subgroup of
, and let be a sequence of consecutive integers. Let
which is a particular kind of weighted restricted sumset. We show that , that if , and also
characterize all sequences of length with . This
result then allows us to characterize when a linear equation
where are
given, has a solution modulo with all
distinct modulo . As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
(where and ) having
distinct terms, for any . Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence
On weighted zero-sum sequences
Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A
be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest
positive integer , denoted by s_A(G), such that any sequence {c_i}_{i=1}^m
with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which
there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0.
When G is a p-group, A contains no multiples of p and any two distinct
elements of A are incongruent mod p, we show that s_A(G) is at most if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is
the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is
essentially best possible.
In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G)
when exp(G) is even, showing that, for finite abelian groups of even exponent
and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends
to the infinity. Combined with a lower bound of
, where with 1<n_1|... |n_r, this determines s_{{1,-1}}(G), for even exponent
groups, up to a small order error term. Our method makes use of the theory of
L-intersecting set systems.
Some additional more specific values and results related to s_{{1,-1}}(G) are
also computed.Comment: 24 pages. Accepted version for publication in Adv. in Appl. Mat
Representation of Finite Abelian Group Elements by Subsequence Sums
Let be a finite and nontrivial
abelian group with . A conjecture of Hamidoune says that if
is a sequence of integers, all but at most one relatively prime
to , and is a sequence over with ,
the maximum multiplicity of at most , and ,
then there exists a nontrivial subgroup such that every element
can be represented as a weighted subsequence sum of the form
, with a subsequence of . We give two
examples showing this does not hold in general, and characterize the
counterexamples for large .
A theorem of Gao, generalizing an older result of Olson, says that if is
a finite abelian group, and is a sequence over with , then either every element of can be represented as a
-term subsequence sum from , or there exists a coset such that
all but at most terms of are from . We establish some very
special cases in a weighted analog of this theorem conjectured by Ordaz and
Quiroz, and some partial conclusions in the remaining cases, which imply a
recent result of Ordaz and Quiroz. This is done, in part, by extending a
weighted setpartition theorem of Grynkiewicz, which we then use to also improve
the previously mentioned result of Gao by showing that the hypothesis can be relaxed to , where
d^*(G)=\Sum_{i=1}^{r}(n_i-1). We also use this method to derive a variation
on Hamidoune's conjecture valid when at least of the are
relatively prime to