Published paper
Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
MSC 2020
11J2011J1311J8337A44
Abstract
In 2019 Kleinbock and Wadleigh proved a “zero-one law” for uniform inhomogeneous Diophantine approximations. We generalize this statement to arbitrary weight functions and establish a new and simple proof of this statement, based on the transference principle. We also give a complete description of the sets of -Dirichlet pairs with a fixed matrix in this set-up from the Lebesgue-measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work, and can sometimes be strengthened, for weighted Diophantine approximations.
Role in dependence graphs
Proof-critical source
Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
This paper is included only for the following marked statements:
- Theorem 1.3 · statement PDF p. 5; proof PDF pp. 24–26Zero–full Lebesgue-measure law for weighted uniform inhomogeneous approximation.
- Lemma D and weighted transference theorems · Lemma D PDF pp. 11–12; Theorem 1.4 statement PDF p. 6 and proof PDF p. 18Quantitative pseudo-compound-body transference used to pass between homogeneous and inhomogeneous approximation statements.
- Theorem 1.5 · statement PDF p. 6; proof PDF pp. 18–21Measure conclusion obtained from a sharp second-moment limsup estimate.
- Theorem 3.1 and dual-body setup · PDF p. 13; definitions and citation discussion PDF pp. 10–11Classical geometry-of-numbers input used in the paper's lattice formulation.
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