In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine-Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira [8].
The Mass Transference Principle proved by Beresnevich and Velani in 2006 is a celebrated and highly influential result which allows us to infer Hausdorff measure statements for lim sup sets of balls in R n from a priori weaker Lebesgue measure statements. The Mass Transference Principle and subsequent generalisations have had a profound impact on several areas of mathematics, especially Diophantine Approximation. In the present paper, we prove a considerably more general form of the Mass Transference Principle which extends known results of this type in several distinct directions. In particular, we establish a mass transference principle for lim sup sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in R n . Furthermore, our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff g-measure statements to Hausdorff f -measure statements. We conclude the paper with an application of our mass transference principle to a general class of random lim sup sets.2000 Mathematics Subject Classification: Primary 11J83, 28A78; Secondary 11K60.
Motivated by a classical question of Mahler (1984), Levesley, Salp, and Velani (2007) showed that the Hausdorff measure of the set of points in the middle-third Cantor set which are ψ-well-approximable by triadic rationals satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of this set is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on ψ.In this article, we prove an analogue of this result, obtaining a zero-full dichotomy for Hausdorff measure, in the setting of more general self-conformal sets with an appropriate adapted notion of approximation. Unlike in the work of Levesley, Salp, and Velani, we show that we are unable to apply the Mass Transference Principle due to Beresnevich and Velani (2006) in our setting. Instead, our proof relies on recasting the problem in the language of symbolic dynamics and appealing to several concepts from thermodynamic formalism, eventually enabling us to use an analogue of the mass distribution principle. In addition to demonstrating how our main result naturally extends the work of Levesley, Salp, and Velani, and complements some recent work of Baker (2018), we apply our main result to obtain a Jarník type statement for the Hausdorff measure of the set of badly approximable numbers which are "well-approximable" in some sense by fixed quadratic irrationals. This relies on the fact that the set of badly approximable numbers with partial quotients bounded above by a fixed M P N forms a self-conformal, but not self-similar, set. Hence this is a novel application of our main result which does not follow directly from previous results in this direction.
In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley, Salp, and Velani (2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.