相关论文: Diophantine Approximation on varieties V: Algebrai…

For $\theta$ a non-algebraic point on a quasi projective variety over a number field, I prove that $\theta$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications…

The metric Bezout Theorem proved in an earlier paper can be extended to a derivative version that compares derivatives of the algebraic distance of a point $\theta$ to two properly intersecting cycles in projective space with the…

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms…

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…

Let $\Theta = (\theta_1,\theta_2,\theta_3)\in \mathbb{R}^3$. Suppose that $1,\theta_1,\theta_2,\theta_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$ \alpha(\Theta) = \sup \{\gamma >0:\,\,\, \limsup_{t\to…

In this paper, we introduce an algebro-geometric formulation for Faltings' theorem on diophantine approximation on abelian varieties using an improvement of Faltings-Wustholz observation over number fields. In fact, we prove that, for any…

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a…

A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models…

This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real…

Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the…

A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $\Theta$, $\Phi$ and…

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the…

We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In…

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta…

We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the…

We generalize Gel'fond's criterion of algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the additive group of complex numbers, instead of…

We prove a number of results on the metric and non-metric theory of Diophantine approximation for Yu's multidimensional variant of Mahler's classification of transcendental numbers. Our results arise as applications of well known results in…

In this paper, we explore several threads arising from our recent joint work on arithmetic holonomy bounds, which were originally devised to prove new irrationality results based on the method of Ap\'ery limits. We propose a new method to…

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…