23,035 research outputs found
Counting MSTD Sets in Finite Abelian Groups
In an abelian group G, a more sums than differences (MSTD) set is a subset A
of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets
in finite abelian groups, extending previous results of Nathanson. The proof
contains an application of a recently resolved conjecture of Alon and Kahn on
the number of independent sets in a regular graph.Comment: 17 page
Some new results about a conjecture by Brian Alspach
In this paper we consider the following conjecture, proposed by Brian
Alspach, concerning partial sums in finite cyclic groups: given a subset of
of size such that ,
it is possible to find an ordering of the elements of
such that the partial sums , , are nonzero
and pairwise distinct. This conjecture is known to be true for subsets of size
in cyclic groups of prime order. Here, we extend such result to any
torsion-free abelian group and, as a consequence, we provide an asymptotic
result in .
We also consider a related conjecture, originally proposed by Ronald Graham:
given a subset of , where is a prime, there
exists an ordering of the elements of such that the partial sums are all
distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based
on the Alon's combinatorial Nullstellensatz, we prove the validity of such
conjecture for subsets of size
A problem on partial sums in abelian groups
In this paper we propose a conjecture concerning partial sums of an arbitrary
finite subset of an abelian group, that naturally arises investigating simple
Heffter systems. Then, we show its connection with related open problems and we
present some results about the validity of these conjectures
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