2,479 research outputs found
Spanning subset sums for finite Abelian groups
AbstractWe survey the state of research to determine the maximum size of a nonspanning subset of a finite abelian group G of order n. The smallest prime factor of n, denote it here by p, plays a crucial role. For prime order, G=Zp, this is essentially an old problem of Erdős and Heilbronn, which can be solved using a result of Dias da Silva and Hamidoune. We provide a simple new proof for the solution when n is even (p=2). For composite odd n, we deduce the solution, for n⩾2p2, from results obtained years ago by Diderrich and, recently, by Gao and Hamidoune. Only a small family of cases remains unsettled
F-sets and finite automata
The classical notion of a k-automatic subset of the natural numbers is here
extended to that of an F-automatic subset of an arbitrary finitely generated
abelian group equipped with an arbitrary endomorphism F. This is
applied to the isotrivial positive characteristic Mordell-Lang context where F
is the Frobenius action on a commutative algebraic group G over a finite field,
and is a finitely generated F-invariant subgroup of G. It is shown
that the F-subsets of introduced by the second author and Scanlon are
F-automatic. It follows that when G is semiabelian and X is a closed subvariety
then X intersect is F-automatic. Derksen's notion of a k-normal subset
of the natural numbers is also here extended to the above abstract setting, and
it is shown that F-subsets are F-normal. In particular, the X intersect
appearing in the Mordell-Lang problem are F-normal. This generalises
Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.Comment: The final section is revised following an error discovered by
Christopher Hawthorne; it is no longer claimed that an F-normal subset has a
finite symmetric difference with an F-subset. The main theorems of the paper
remain unchange
The Freiman--Ruzsa Theorem over Finite Fields
Let G be a finite abelian group of torsion r and let A be a subset of G. The
Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a
coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by
Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute
constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent
works, which have, in fact, yielded a tight bound. In this work, we establish
the same conjecture for any prime torsion
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