Related papers: A note on zero-one laws in metrical Diophantine ap…
We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.
We address the problem of the continuum limit for a system of Hausdorff lattices (namely lattices of isolated points) approximating a topological space $M$. The correct framework is that of projective systems. The projective limit is a…
We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.
Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation,…
We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the…
This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real…
For two properly intersecting effective cycles in projective space X,Y, and their intersection product Z, the metric Bezout Theorem relates the degrees, heights of X,Y, and Z, as well as their distances and algebraic distances to a given…
In this paper, we consider the problem of counting Diophantine inequalities with multiple natural constraints. We prove a very general result in this setting using dynamical techniques. More precisely, we consider the joint asymptotic…
In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set $\mathcal{K}$, i.e. approximating the points in $\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K. Mahler. By using…
A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a…
We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric…
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic…
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the…
We prove an easy statement about inhomogeneous approximation in metric theory of Diophantine Approximation.
We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives…
We prove a result on the existence of linear forms of a given Diophantine type.
The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss…
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…
In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower…
We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hat{\nu},$ we calculate the Hausdorff dimension of the…