Related papers: Metric bootstraps for limsup sets
The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong…
We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\Bbb R^n$. The correspondence between multidimensional…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different…
This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of…
By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by…
Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable'…
Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…
In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of $m\times n$ matrices over $\Bbb R$ is…
In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine…
In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine-Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on…
We calculate the measure and Hausdorff dimension of sets of matrices over fields of formal power series with good approximation properties for a restricted set of denominators.
We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical…
It is shown that for any translation invariant outer measure M, the M-measure of the intersection of any subset of R^n that is invariant under rational translations and which does not have full Lebesgue measure with an the closure of an…
The method of variational bootstrapping, based on canonical variational completion, allows one to construct a Lagrangian for a physical theory depending on two sets of field variables, starting from a guess of the field equations for only…
Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of…
Consider $M$-estimation in a semiparametric model that is characterized by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. As a general purpose approach to statistical inferences, the bootstrap has found…
Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $\Psi$-well approximable sets. These foundational results have since been generalised to the…
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long standing problem of the existence of…
The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine…