Related papers: Rational approximation to algebraic varieties and …

Let A be an abelian variety defined over a number field K, the number of torsion points rational over a finite extension L is bounded polynomially in terms of the degree [L : K]. When A is isogenous to a product of simple abelian varieties…

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of…

We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

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 introduce a class of real algebraic varieties characterised by a simple rationality condition, which exhibit strong properties regarding approximation of continuous and smooth mappings by regular ones. They form a natural counterpart to…

We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $\omega_2(\xi)$, $\widehat{\omega}_2(\xi)$, $\lambda_2(\xi)$,…

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable…

This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a…

Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for…

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We then present a simple application, related to possible correlations between trace…

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the…

The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each…

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical…

We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.

Let $E\subset [0,1)^{d}$ be a set supporting a probability measure $\mu$ with Fourier decay $|\widehat{\mu}({\bf{t}})|\ll (\log |{\bf{t}}|)^{-s}$ for some constant $s>d+1.$ Consider a sequence of expanding integral matrices…

The computational complexity of reasoning within the Dempster-Shafer theory of evidence is one of the main points of criticism this formalism has to face. To overcome this difficulty various approximation algorithms have been suggested that…

We prove a new lower bound for the exponent of growth of the best two-dimensional Diophantine approximations with respect to Euclidean norm.

We prove that the existence of finite combinatorial objects such as affine planes, mutually orthogonal Latin squares, and resolvable balanced incomplete block designs can be reformulated as the existence of certain algorithmic reductions…