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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
Singer quadrangles
[no abstract available
Direct sums and products in topological groups and vector spaces
We call a subset of an abelian topological group : (i)
provided that for every open neighbourhood of one
can find a finite set such that the subgroup generated by
is contained in ; (ii) if, for every
family of integer numbers, there exists such that the
net \left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}
converges to ; (iii) provided that and for every neighbourhood of there exists a neighbourhood of
such that, for every finite set and each set of integers, implies that for all
. We prove that: (1) an abelian topological group contains a direct
product (direct sum) of -many non-trivial topological groups if and
only if it contains a topologically independent, absolutely (Cauchy) summable
subset of cardinality ; (2) a topological vector space contains
as its subspace if and only if it has an infinite
absolutely Cauchy summable set; (3) a topological vector space contains
as its subspace if and only if it has an
multiplier convergent series of non-zero elements.
We answer a question of Hu\v{s}ek and generalize results by
Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki
Counting abelian varieties over finite fields via Frobenius densities
Let be a principally polarized abelian variety over a finite
field with commutative endomorphism ring; further suppose that either is
ordinary or the field is prime. Motivated by an equidistribution heuristic, we
introduce a factor for each place of , and
show that the product of these factors essentially computes the size of the
isogeny class of .
The derivation of this mass formula depends on a formula of Kottwitz and on
analysis of measures on the group of symplectic similitudes, and in particular
does not rely on a calculation of class numbers.Comment: Main text by Achter, Altug and Gordon; appendix by Li and Ru
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