24 research outputs found
Metrical Diophantine approximation for quaternions
Analogues of the classical theorems of Khintchine, Jarnik and
Jarnik-Besicovitch in the metrical theory of Diophantine approximation are
established for quaternions by applying results on the measure of general `lim
sup' sets.Comment: 30 pages. Some minor improvement
Diophantine approximation in Banach spaces
In this paper, we extend the theory of simultaneous Diophantine approximation
to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very
general framework and define what it means for such a theorem to be optimal. We
show that optimality is implied by but does not imply the existence of badly
approximable points
Weighted approximation for limsup sets
Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine
approximation are fundamental results on the metric properties of -well
approximable sets. These foundational results have since been generalised to
the framework of weighted Diophantine approximation for systems of real linear
forms (matrices). In this article, we prove analogues of these weighted results
in a range of settings including the -adics (Theorems 7 and 8), complex
numbers (Theorems 9 and 10), quaternions (Theorems 11 and 12), and formal power
series (Theorems 13 and 14). We also consider approximation by uniformly
distributed sequences. Under some assumptions on the approximation functions,
we prove a 0-1 dichotomy law (Theorem 15). We obtain divergence results for any
approximation function under some natural restrictions on the discrepancy
(Theorems 16, 17, and 19).
The key tools in proving the main parts of these results are the weighted
ubiquitous systems and weighted mass transference principle introduced recently
by Kleinbock and Wang [Adv. Math. 428 (2023), Paper No. 109154], and Wang and
Wu [Math. Ann. 381 (2021), no. 1-2, 243--317] respectively.Comment: 73 pages, comments welcom
Diophantine approximation in Banach spaces
In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life
Mathematical source references
This list of references is intended to be a convenient reference source for those interested in the historical origin of common mathematical ideas, The topics mentioned are mostly those met in a degree course in mathematics. For each entry the list attempts to give an exact source reference with comments about priority. There are now available other historical reference sources for mathematics on the internet but with a different style of presentation.<br/
Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
Our monograph presents the foundations of the theory of groups and semigroups
acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and
extends a long list of results by many authors. We make it a point to avoid any
assumption of properness/compactness, keeping in mind the motivating example of
, the infinite-dimensional rank-one symmetric space of
noncompact type over the reals. The monograph provides a number of examples of
groups acting on which exhibit a wide range of phenomena not
to be found in the finite-dimensional theory. Such examples often demonstrate
the optimality of our theorems. We introduce a modification of the Poincar\'e
exponent, an invariant of a group which gives more information than the usual
Poincar\'e exponent, which we then use to vastly generalize the Bishop--Jones
theorem relating the Hausdorff dimension of the radial limit set to the
Poincar\'e exponent of the underlying semigroup. We give some examples based on
our results which illustrate the connection between Hausdorff dimension and
various notions of discreteness which show up in non-proper settings. We
construct Patterson--Sullivan measures for groups of divergence type without
any compactness assumption. This is carried out by first constructing such
measures on the Samuel--Smirnov compactification of the bordification of the
underlying hyperbolic space, and then showing that the measures are supported
on the bordification. We study quasiconformal measures of geometrically finite
groups in terms of (a) doubling and (b) exact dimensionality. Our analysis
characterizes exact dimensionality in terms of Diophantine approximation on the
boundary. We demonstrate that some Patterson--Sullivan measures are neither
doubling nor exact dimensional, and some are exact dimensional but not
doubling, but all doubling measures are exact dimensional.Comment: A previous version of this document included Section 12.5 (Tukia's
isomorphism theorem). The results of that subsection have been split off into
a new document which is available at arXiv:1508.0696