Formes linéaires invariantes par translation et approximation rationnelle
Hausdorff dimension and a generalized form of simultaneous Diophantine approximation
Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation.
Inhomogeneous Diophantine approximation on integer polynomials with non-monotonic error function
We prove that the Lebesgue measure of the set of real points which are inhomogeneously Ψ-approximable by polynomials, where Ψ is not necessarily monotonic, is zero.
Introduction to Liouville Numbers
The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and [...] It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is...
Khintchine types of translated coordinate hyperplanes
There has been great interest in developing a theory of "Khintchine types" for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes...
Khintchine-type theorems on manifolds
Kronecker-type sequences and nonarchimedean diophantine approximations
Les suites eutaxiques
Meilleures approximations diophantiennes d’un élément du tore
Meilleures approximations d'un élément du tore 𝕋² et géométrie de la suite des multiples de cet élément
Metric diophantine approximation in Julia sets of expanding rational maps
Metric Diophantine approximation with two restricted variables. IV. Miscellaneous results
Metric inhomogeneous Diophantine approximation on the field of formal Laurent series
Metric properties of alternating Lüroth series
Metric results on the approximation of zero by linear combinations of independent and of dependent rationals.
Metric theorems on the approximation of zero by a linear combination of polynomials with integral coefficients
Metric theorems on the estimation of integrals by sums
Metrical theorems for inhomogeneous Diophantine approximation in positive characteristic
Multiplicative zero-one laws and metric number theory
We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete 'multiplicative' zero-one law is established akin to the 'simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile 'cross fibering principle'. In a nutshell it enables us to 'lift' zero-one laws to higher...