272 research outputs found
Inverse results for weighted Harborth constants
For a finite abelian group the Harborth constant is defined as the
smallest integer such that each squarefree sequence over of length
has a subsequence of length equal to the exponent of whose terms sum
to . The plus-minus weighted Harborth constant is defined in the same way
except that the existence of a plus-minus weighted subsum equaling is
required, that is, when forming the sum one can chose a sign for each term. The
inverse problem associated to these constants is the problem of determining the
structure of squarefree sequences of maximal length that do not yet have such a
zero-subsum. We solve the inverse problems associated to these constant for
certain groups, in particular for groups that are the direct sum of a cyclic
group and a group of order two. Moreover, we obtain some results for the
plus-minus weighted Erd\H{o}s--Ginzburg--Ziv constant
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
On products of k atoms II
Let be a Krull monoid with class group such that every class contains
a prime divisor (for example, rings of integers in algebraic number fields or
holomorphy rings in algebraic function fields). For , let
denote the set of all with the following
property: There exist atoms such that . Furthermore, let and . The sets
are intervals which are finite if and only
if is finite. Their minima can be expressed in terms of
. The invariants depend only on the class group ,
and in the present paper they are studied with new methods from Additive
Combinatorics
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