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Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3,3^{2h+1})
Using a class of permutation polynomials of obtained from the
Ree-Tits symplectic spreads in , we construct a family of skew
Hadamard difference sets in the additive group of . With the help
of a computer, we show that these skew Hadamard difference sets are new when
and . We conjecture that they are always new when .
Furthermore, we present a variation of the classical construction of the twin
prime power difference sets, and show that inequivalent skew Hadamard
difference sets lead to inequivalent difference sets with twin prime power
parameters.Comment: 18 page
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
Difference sets and frequently hypercyclic weighted shifts
We solve several problems on frequently hypercyclic operators. Firstly, we
characterize frequently hypercyclic weighted shifts on ,
. Our method uses properties of the difference set of a set with
positive upper density. Secondly, we show that there exists an operator which
is -frequently hypercyclic, yet not frequently hypercyclic and that
there exists an operator which is frequently hypercyclic, yet not
distributionally chaotic. These (surprizing) counterexamples are given by
weighted shifts on . The construction of these shifts lies on the
construction of sets of positive integers whose difference sets have very
specific properties
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