1,492 research outputs found
Subsets of finite groups exhibiting additive regularity
In this article we aim to develop from first principles a theory of sum sets
and partial sum sets, which are defined analogously to difference sets and
partial difference sets. We obtain non-existence results and characterisations.
In particular, we show that any sum set must exhibit higher-order regularity
and that an abelian sum set is necessarily a reversible difference set. We next
develop several general construction techniques under the hypothesis that the
over-riding group contains a normal subgroup of order 2. Finally, by exploiting
properties of dihedral groups and Frobenius groups, several infinite classes of
sum sets and partial sum sets are introduced
Pseudofinite structures and simplicity
We explore a notion of pseudofinite dimension, introduced by Hrushovski and
Wagner, on an infinite ultraproduct of finite structures. Certain conditions on
pseudofinite dimension are identified that guarantee simplicity or
supersimplicity of the underlying theory, and that a drop in pseudofinite
dimension is equivalent to forking. Under a suitable assumption, a
measure-theoretic condition is shown to be equivalent to local stability. Many
examples are explored, including vector spaces over finite fields viewed as
2-sorted finite structures, and homocyclic groups. Connections are made to
products of sets in finite groups, in particular to word maps, and a
generalization of Tao's algebraic regularity lemma is noted
- …