Related papers: Multiplicative Diophantine Exponents of Hyperplane…
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…
We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning…
We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.
We study some problems in metric Diophantine approximation over local fields of positive characteristic.
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…
In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that…
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…
We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.
We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We…
We construct a class of multiple Legendre polynomials and prove that they satisfy an Ap\'ery-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree.…
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.
This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in…
We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\,…
The input to the distant representatives problem is a set of $n$ objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are…
We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays…
This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of…
We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related to geometrical properties of the classical Markov and Lagrange spectra…
In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called exponents. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes…