Exponents of Diophantine approximation

关于格点的丢番图指数谱

On the spectrum of Diophantine exponents of lattices

本文研究了任意维数格点的弱均匀丢番图指数的谱特性。

Diophantine Exponents of Affine Subspaces: The Simultaneous Approximation Case

We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.

Diophantine exponents of lattices

In this paper we define Diophantine exponents of lattices and investigate some of their properties. We prove transference inequalities and construct some examples with the help of Schmidt's subspace theorem.

Multiplicative Diophantine exponents of hyperplanes and their nondegenerate submanifolds

We consider multiparameter dynamics on the space of unimolular lattices. Along with quantitative nondivergence we prove that multiplicative Diophantine exponents of hyperplanes are inherited by their nondegenerate submanifolds.

关于格的均匀丢番图指数

On uniform Diophantine exponents of lattices

本文研究格的弱均匀丢番图指数谱，并在二维情形下给出了该谱的完整刻画。

Linear forms of a given Diophantine type and lattice exponents

In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.

关于格点的丢番图指数

On Diophantine exponents of lattices

本文研究了d维格点的丢番图指数，通过归约到二维情况并应用度量理论方法，揭示了其指数谱特性。

关于实数的弱一致丢番图指数

On the weak uniform Diophantine exponent of a real number

本文提出实数的弱一致丢番图指数新概念，并完整刻画了该指数的所有可能取值谱。

Exponents of inhomogeneous Diophantine Approximation

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 the exponent of approximation to a generic point in R^n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual forms.

关于格的一致丢番图指数的平凡性

On triviality of uniform Diophantine exponents of lattices

本文研究格的一致丢番图指数，证明其仅能取得平凡值。

Intermediate Diophantine exponents and parametric geometry of numbers

This is a revised compilation of the papers arXiv:1105.1554 and arXiv:1105.5823. We develop some of the ideas belonging to W.Schmidt and L.Summerer to define intermediate Diophantine exponents and split several transference inequalities into a chain of inequalities for intermediate exponents.

Transference inequalities for multiplicative Diophantine exponents

In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.

On Diophantine exponents and Khintchine's transference principle

In this paper we improve estimates of Jarnik and Apfelbeck for uniform Diophantine exponents of transposed systems of linear forms and generalize to the case of an arbitrary system the estimates of Laurent and Bugeaud for individual exponents. The method proposed also gives a better constant in Mahler's transference theorem.

An inequality for Diophantine exponents in the case of simultaneous approximation

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.

On a Problem in Diophantine Approximation

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

Remarks on Diophantine approximation in function fields

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

Diophantine approximation on lines with prime constraints

We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.

On Harrap's conjecture in Diophantine approximation

We prove a conjecture due to Stephen Harrap on inhomogeneous linear Diophantine approximation related to ${\rm BAD}(α,β)$ sets.

An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation

We provide a lower bound for the ratio between the ordinary and uniform exponent of both simultaneous Diophantine approximation and Diophantine approximation by linear forms in any dimension. This lower bound was conjectured by Schmidt and Summerer and already shown in dimension $2$ and $3$. This lower bound is reached at regular systems presented in the context of parametric geometry of numbers, and thus optimal.

Note on the spectrum of classical and uniform exponents of Diophantine approximation

Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we establish that the spectrum of the $2n$ exponents of Diophantine approximation in dimension $n\geq3$ is a subset of $\mathbb{R}^{2n}$ with non empty interior.

A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.

The best Diophantine approximations: the phenomenon of degenerate dimension

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 Diophantine approximations. Originally most of these results have been established by the author in few years ago. Here we collect all of them together and give some new formulations.

Diophantine approximations with positive integers: some remarks

We give some comments on W.M. Schmidt's theorem on Diophantine approximations with positive integers and our recent results on the topic.

Restricted simultaneous Diophantine approximation

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.

Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.

针对最后一个角度的具有已知丢番图指数的子空间的构造

Construction of subspaces with known Diophantine exponents for the last angle

本文利用数的几何方法，构造了 $\R^n$ 中能够计算最后一个角度关联丢番图指数的线性子空间，推广了实数的无理性测度。

Simultaneous two-dimensional best Diophantine approximations in the Euclidean norm

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

A note on weighted simultaneous Diophantine approximation on manifolds

In this note, we present an improvement to a recent result due to Beresnevich, Levesley, and Ward (2021) pertaining to weighted simultaneous Diophantine approximation on manifolds.

Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation

We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.

Distribution modulo one and diophantine approximation by Yann Bugeaud

Review of the book: Distribution modulo one and Diophantine approximation, by Yann Bugeaud, Cambridge University Press 2012. ISBN 978-0521111690, 316 pp.

与线性子空间相关的丢番图指数的独立性

Independence of the Diophantine exponents associated with linear subspaces

本文研究Schmidt提出的丢番图逼近推广问题，证明线性子空间丢番图指数相关函数间无光滑关系，并构造可计算指数的子空间。

Intersective polynomials and Diophantine approximation, II

By applying Schmidt's lattice method, we prove results on simultaneous Diophantine approximation modulo 1 for systems of polynomials in a single prime variable provided that certain local conditions are met.

弱逼近、迪奥菲安指数和二维格点

Weak approximations, Diophantine exponents and two-dimensional lattices

本文研究二维格点的迪奥菲安指数及其相关弱统一逼近性质，利用连分数理论分析普通与统一迪奥菲安指数间的不等式关系，揭示两个实数无理数度量函数的行为关联。

Best Simultaneous Diophantine Approximations under a Constraint on the Denominator

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximation results are applied to the decomposition of integer vectors.

Best simultaneous Diophantine approximations for some pairs of algebraic numbers

We compute the sequence of best Diophantine approximations for some pairs of cubic Pisot numbers which do not satisfy the Property (F).

Counting intrinsic Diophantine approximations in simple algebraic groups

We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.

Transference theorems for Diophantine approximation with weights

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.

Diophantine exponents of lattices and growth of multidimensional analogues of partial quotients

In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.

Diophantine approximation and automorphic spectrum

The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.

Exponents of Diophantine Approximation and Sturmian Continued Fractions

Let x be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w_n(x) and w_n^*(x) defined by Mahler and Koksma. We calculate their six values when n=2 and x is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction x by quadratic surds.

Diophantine exponents for systems of linear forms in two variables

We improve on Jarn\'ık's inequality between uniform Diophantine exponent $α$ and ordinary Diophantine exponent $β$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'ık (1949, 1954) proved that $β\ge α(α-1)$. In the present paper we give a better bound in the case $α>1$. We prove that β\ge 1/2(α^2-α+1+\sqrt{(α^2-α+1)^2 +4α^2(α-1)}) if 1\le α\le 2 1/2(α^2-1+\sqrt{(α^2-1)^2+4α(α-1)}) if α\ge 2

Diophantine Approximation Groups, Kronecker Foliations and Independence

We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of $K$-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geometric (graph) rigidities -- of the Theorems of Baker and Lindemann-Weierstrass, the Logarithm Conjecture and the Schanuel Conjecture. There is an Appendix describing diophantine approximation groups as model theoretic types.

On some open problems in Diophantine approximation

We discuss several open problems in Diophantine approximation. Among them there are famous Littlewood's and Zaremba's conjectures as well as some new and not so famous problems.

Theorem on best Diophantine approximations for linear forms

We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.

Metric Diophantine Approximation: aspects of recent work

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'ık, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.

固定矩阵下的非齐次丢番图逼近度量理论

Metric theory of inhomogeneous Diophantine approximations with a fixed matrix

本文针对固定矩阵情形，利用转移原理建立矩阵向量配对与转置矩阵丢番图性质的双向关联，证明齐次丢番图逼近经典结论的对应版本，解答相关公开问题，构建了对应的非齐次丢番图逼近度量理论。

Diophantine approximation with arithmetic functions, I

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

Metrical Diophantine approximation for quaternions

Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.

Zero-one laws in simultaneous and multiplicative Diophantine approximation

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.

On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $α_1x_1+α_2x_2 +y$ with {\it positive} integers $x_1,x_2$.

Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents

We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.

Diophantine approximation exponents on homogeneous varieties

Recent years have seen very important developments at the interface of Diophantine approximation and homogeneous dynamics. In the first part of the paper we give a brief exposition of a dictionary developed by Dani and Kleinbock-Margulis which relates Diophantine properties of vectors to distribution of orbits of flows on the space of unimodular lattices. In the second part of the paper we briefly describe an extension of this dictionary recently developed by the authors, which establishes an analogous dynamical correspondence for general lattice orbits on homogeneous spaces. We concentrate sp…

Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds

The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.

Some remarks on diophantine equations and diophantine approximation

We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various Diophantine equations. We also give many equivalent statements of Mahler's generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue--Mahler for degree $3$ implies the theorem of Thue--Mahler for arbitrary degree $\ge3$, and we relate it with a theorem of Siegel on the rational integral points of the projective line $¶^1…