161 research outputs found
Ultrametric Logarithm Laws, II
We prove positive characteristic versions of the logarithm laws of Sullivan
and Kleinbock-Margulis and obtain related results in Metric Diophantine
Approximation.Comment: submitted to Montasefte Fur Mathemati
Klein polyhedra and lattices with positive norm minima
A Klein polyhedron is defined as the convex hull of nonzero lattice points
inside an orthant of . It generalizes the concept of continued fraction.
In this paper facets and edge stars of vertices of a Klein polyhedron are
considered as multidimensional analogs of partial quotients and quantitative
characteristics of these ``partial quotients'', so called determinants, are
defined. It is proved that the facets of all the Klein polyhedra
generated by a lattice \La have uniformly bounded determinants if and only if
the facets and the edge stars of the vertices of the Klein polyhedron generated
by \La and related to the positive orthant have uniformly bounded
determinants.Comment: 12 pages, 18 reference
Ultrametric Logarithm Laws I
We announce ultrametric analogues of the results of Kleinbock-Margulis for
shrinking target properties of semisimple group actions on symmetric spaces.
The main applications are S-arithmetic Diophantine approximation results and
logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on
trees.Comment: This announcement has been completely revised to reflect many new
developments. Please direct all references to this NEW announcement. It is
now co-authored work. Submitted to Discrete and Continuous Dynamical System
Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows
We show an analogue of a theorem of An, Ghosh, Guan, and Ly on weighted badly
approximable vectors for totally imaginary number fields. We show that for
and
a lattice subgroup, the points of with bounded orbits
under a one-parameter Ad-semisimple subgroup of form a
hyperplane-absolute-winning set. As an application, we also provide a
generalization of a result of Esdahl-Schou and Kristensen about the set of
badly approximable complex numbers.Comment: 17 pages, v2: minor corrections, some statements strengthene
Jarnik-type Inequalities
It is well known due to Jarnik that the set Bad of badly approximable numbers
is of Hausdorff-dimension one. If Bad(c) denotes the subset of x in Bad for
which the approximation constant c > c(x), then Jarnik was in fact more precise
and gave nontrivial lower and upper bounds of the Hausdorff-dimension of Bad(c)
in terms of the parameter c > 0. Our aim is to determine simple conditions on a
framework which allow to extend 'Jarnik's inequality' to further examples;
among the applications, we discuss the set of badly approximable vectors in
with weights and the set of geodesics in the hyperbolic space which avoid a
suitable collection of convex sets.Comment: Comments are welcome! Corrections and modifications in new versio
Domains via approximation operators
In this paper, we tailor-make new approximation operators inspired by rough
set theory and specially suited for domain theory. Our approximation operators
offer a fresh perspective to existing concepts and results in domain theory,
but also reveal ways to establishing novel domain-theoretic results. For
instance, (1) the well-known interpolation property of the way-below relation
on a continuous poset is equivalent to the idempotence of a certain
set-operator; (2) the continuity of a poset can be characterized by the
coincidence of the Scott closure operator and the upper approximation operator
induced by the way below relation; (3) meet-continuity can be established from
a certain property of the topological closure operator. Additionally, we show
how, to each approximating relation, an associated order-compatible topology
can be defined in such a way that for the case of a continuous poset the
topology associated to the way-below relation is exactly the Scott topology. A
preliminary investigation is carried out on this new topology.Comment: 17 pages; 1figure, Domains XII Worksho
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