153 research outputs found
Iterated Sumsets and Subsequence Sums
Let
be a finite abelian group with . The Kemperman
Structure Theorem characterizes all subsets satisfying
and has been extended to cover the case when . Utilizing these results, we provide a precise structural description
of all finite subsets with when
(also when is infinite), in which case many of the pathological
possibilities from the case vanish, particularly for large . The structural description is combined with other arguments to
generalize a subsequence sum result of Olson asserting that a sequence of
terms from having length must either have every element of
representable as a sum of -terms from or else have all but
of its terms lying in a common -coset for some . We show
that the much weaker hypothesis suffices to obtain a
nearly identical conclusion, where for the case is trivial we must allow
all but terms of to be from the same -coset. The bound on
is improved for several classes of groups , yielding optimal lower
bounds for . We also generalize Olson's result for -term subsums to
an analogous one for -term subsums when , with the bound
likewise improved for several special classes of groups. This improves previous
generalizations of Olson's result, with the bounds for optimal.Comment: Revised version, with results reworded to appear less technica
On a Conjecture of Hamidoune for Subsequence Sums
Let G be an abelian group of order m, let S be a sequence of terms from G with k distinct terms, let m ∧ S denote the set of all elements that are a sum of some m-term subsequence of S, and let |S| be the length of S. We show that if |S| ≥ m + 1, and if the multiplicity of each term of S is at most m − k + 2, then either |m ∧ S| ≥ min{m, |S| − m + k − 1}, or there exists a proper, nontrivial subgroup Ha of index a, such that m ∧ S is a union of Ha-cosets, Ha ⊆ m ∧ S, and all but e terms of S are from the same Ha-coset, where e ≤ min{|S|−m+k−2 |Ha| − 1, a − 2} and |m ∧ S| ≥ (e + 1)|Ha|. This confirms a conjecture of Y. O. Hamidoune
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 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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