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

On subgroups of semi-abelian varieties defined by difference equations

Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics onto (T,M) arises from a finite isogeny T \to T. A similar and more general statement is shown for Abelian and semi-abelian varieties. In model-theoretic terms, our result says: Working in an existentially closed difference field, we consider a definable subgroup B of a semi-abelian variety A; assume B does not have a subgroup isogenous to A'(F) for some twisted fixed field F, and some semi-Abelian variety A'. Then B with the induced structure is stable and stably embedded. This implies in particular that for any n>0, any definable subset of B^n is a Boolean combination of cosets of definable subgroups of B^n. This result was already known in characteristic 0 where indeed it holds for all commutative algebraic groups ([CH]). In positive characteristic, the restriction to semi-abelian varieties is necessary.Comment: Revised version, to appea