Mass transference principle from rectangles to rectangles in Diophantine approximation

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in m m -dimensional Euclidean space, or a set of n n -by- m m systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in m m -dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of n n -by- m m systems of linear forms. Consequently, a class of m m -dimensional results in Diophantine approximation can be bootstrapped to corresponding n n -by- m m -dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

We present a comprehensive framework for the study of the size and large intersection properties of limsup sets that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of Lévy processes.

Given and , let denote the classical set of -approximable points in , which consists of that lie within distance from the lattice for infinitely many . In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold of and any almost all points on are not -approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set . In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], which enables us to find a sharp lower bound for for any submanifold of and any satisfying . Here is the codimension of . We also show that the condition on is best possible and extend the result to general approximating functions.

Let $\beta\in(1,2)$ and $x\in [0,1/(\beta-1)]$. We call a sequence $(\epsilon_{i})_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ a $\beta$-expansion for $x$ if $x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}$. We call a finite sequence $(\epsilon_{i})_{i=1}^{n}\in\

Uniformly Approximable Numbers and the Uniform Approximation Spectrum

On Dirichlet's theorem concerning diophantine approximation

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarník and Besicovitch regarding well-approximable and badly-approximable points.In this paper we consider three inhomogeneous problems that further develop these classical results.Firstly, we obtain a Jarník type theorem for the set S × 2 (ψ; θ) of multiplicatively approximable points in the plane R 2 .This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting (θ = 0).Next, we establish a Jarník type theorem for the set S × 2 (ψ; θ) ∩ C where C is a non-degenerate planar curve.This completes the Hausdorff theory for planar curves and clarifies a potential oversight in [2].Finally, we show that the set Bad(i, j; θ) of simultaneously inhomogeneously badly approximable points in R 2 is of full dimension.The underlying philosophy behind the proof has other applications; e.g.towards establishing the inhomogeneous version of Schmidt's Conjecture.The higher dimensional analogues of the planar results are also discussed.

In this paper the metric theory of Diophantine approximation of linear forms that are of mixed type is investigated. Khintchine–Groshev theorems are established together with the Hausdorff measure generalizations. The latter includes the original dimension results obtained in [H. Dickinson, The Hausdorff dimension of sets arising in metric Diophantine approximation, Acta Arith.68(2) (1994) 133–140] as special cases.

A dimensional mass transference principle from ball to rectangles for projections of Gibbs measures and applications

Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of Ψ-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 p -adics (theorems 7 and 8), complex numbers (theorems 9 and 10), quaternions (theorems 11 and 12), and formal power series (theorems 13 and 14). 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–Wang (2023 Adv. Math. ) and Wang–Wu (2021 Math. Ann ).

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space.