Michel Laurent
Let $Θ$ be a point in ${\bf R}^n$. We split the classical Khintchine's Transference Principle into $n-1$ intermediate estimates which connect exponents $ω_d(Θ)$ measuring the sharpness of the approximation to $Θ$ by linear rational varieties of dimension $d$, for $0\le d \le n-1$. We also review old and recent results related to these $n$ exponents.
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Using the Davenport-Heilbronn circle method, we show that for almost all additive Diophantine inequalities of degree $k$ in more than $2k$ variables the expected asymptotic formula for the density of solutions holds true. This appears to be the first metric result on Diophantine inequalities.
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.
In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems, including Ingham's exponential theorem as well as some new results. Diophantine inequalities and their connection with Cramer's and Andrica's conjectures are also discussed.
A note on some Diophantine inequalities over adelic curves
本文在不假设Northcott性质的条件下，为proper adelic曲线上一类作为Roth不等式加强版的丢番图不等式组建立了“大解”数量的上界。
The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.
We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be optimal. However, they may be sharpened by taking into account two further uniform exponents.
The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many solutions in primes exist.
Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.
Denote by $s_0^{(r)}$ the least integer such that if $s \ge s_0^{(r)}$, and $F$ is a cubic form with real coefficients in $s$ variables that splits into $r$ parts, then $F$ takes arbitrarily small values at nonzero integral points. We bound $s_0^{(r)}$ for $r \le 6$.
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.
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.
Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We study this problem on average over \alp_3 and generalize previous work of Bourgain on quadratic ternary diagonal forms to general degree k.
We apply Freeman's variant of the Davenport-Heilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a positive number c.
Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$ Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3,\,p_4,\,p_5,\,p_6$ for sufficiently large real number $N$.
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.
We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit equality for suitable $\ux$, which turns out rather surprising. For $n=2$ our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved $\ux$, and in special cases we can show they are sharp. Proofs are based on the variational principle in parametric geometry of numbers, we enclose sketches of …
Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N \end{equation*} is solvable in primes $p_1,p_2,p_3$. Moreover, we also prove that the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5,p_6$, which improves the previous result $1<c<\frac{37}{18},c\neq2$.
Consider a form $g(x_1,...,x_s)$ of degree $d$, having coefficients in the completion $F_q((1/t))$ of the field of fractions $F_q(t)$ associated to the finite field $F_q$. We establish that whenever $s>d^2$, then the form $g$ takes arbitrarily small values for non-zero arguments $x\in F_q[t]^s$. We provide related results for problems involving distribution modulo $F_q[t]$, and analogous conclusions for quasi-algebraically closed fields in general.
This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ θ,$ where $ θ>2$ is real and non-integral. For fixed non-zero real numbers $ λ_i $ not all of the same sign we write \begin{equation*} \mathcal F (\textbf{x}) = λ_1 x_1^θ+ \cdots + λ_s x_s^θ. \end{equation*} For a fixed positive real number $ τ$ we give an asymptotic formula for the number of positive integer solutions of the inequality $ | \mathcal F (\textbf{x}) | < τ$ inside a box of side length $P.$ Moreover, we investigate the problem of representing a large positive real number by a positiv…
We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.
This paper is a sequel to our previous paper arXiv:1105.1554, where we defined two types of intermediate Diophantine exponents, connected them to Schmidt exponents and split Dyson's transference inequality into a chain of inequalities for intermediate exponents. Here we present splitting of some other transference inequalities involving both regular and uniform Diophantine exponents.
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.
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.
A New Diophantine Approximation Inequality on Surfaces and Its Applications
本文证明了曲面上闭子概形的丢番图逼近不等式，推广了Ru-Vojta和Heier-Levin的近期结果，并应用于仿射曲面上的整点退化、最大公因数不等式及S-单位方程，同时在复分析背景下建立了第二主定理型不等式。
We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector $\underlineξ=(ξ,ξ^{2},\ldots,ξ^{N})$, in various dimensions $N$. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with $\mathbb{Q}$-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, …
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.
A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessar…
Diophantine transference principle over function fields
本文研究函数域上的丢番图转移原理，通过适配 Beresnevich 和 Velani 的方法，将齐次丢番图逼近结果推广至非齐次情形，导出非齐次 Baker-Sprindzuk 猜想，并证明了一般非极值情形的上界。
In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.
Khintchine's and Dyson's transference theorems can be very easily deduced from Mahler's transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler's theorem. Some extra considerations are required, for instance, induction by the dimension. In this paper we propose an analogue of Mahler's theorem which implies the multiplicative transference theorem immediately.
Let $F_{n}$ be the $n$-th Fibonacci number. Put $φ=\frac{1+\sqrt5}{2}$. We prove that the following inequalities hold for any real $α$: 1) $\inf_{n \in \mathbb N} ||F_nα||\le\frac{φ-1}{φ+2}$, 2) $\liminf_{n\to \infty}||F_nα||\le 1/5$, 3) $\liminf_{n \to \infty}||φ^n α||\le 1/5$. These results are the best possible.
By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by shrinking these rectangles to smaller rectangles has full Hausdorff measure or to say transfer a full measure property to a full Hausdorff measure property for brevity. The limsup sets generated by balls or generated by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Mi…
Moment transference principles and multiplicative diophantine approximation on hypersurfaces
该研究确定了超曲面上的一般乘法丢番图逼近速率，分析了四种不同情形，并利用傅里叶空间中的几何与算术方法建立了矩转移原理通用框架，实现了勒贝格数据到其他测度数据的转换。
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。
On a system of two Diophantine inequalities with six prime variables
本文研究了在特定条件下，两个丢番图不等式系统在素数变量下可解的问题，改进了Han-Liu-Zhang的结果。
The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M.Schmidt which counts the number of approximates to Diophantine inequalit…
In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the i…
Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. …
Let $Θ=(α,β)$ be a point in $\bR^2$, with $1,α,β$ linearly independent over $\bQ$. We attach to $Θ$ a quadruple $Ω(Θ)$ of exponents which measure the quality of approximation to $Θ$ both by rational points and by rational lines. The two ``uniform'' components of $Ω(Θ)$ are related by an equation, due to Jarn{\'ı}k, and the four exponents satisfy two inequalities which refine Khintchine's transference principle. Conversely, we show that for any quadruple $Ω$ fulfilling these necessary conditions, there exists a point $Θ\in \bR^2$ for which $Ω(Θ) =Ω$.
We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.
We find new inequalities between uniform and individual Diophantine exponents for three-dimensional Diophantine approximations. Also we give a result for two linear forms in two variables. The results improves V.Jarnik's theorem (1954).
In this paper we consider a multiparametric version of Wolfgang Schmidt and Leonard Summerer's parametric geometry of numbers. We apply this approach in two settings: the first one concerns weighted Diophantine approximation, the second one concerns Diophantine exponents of lattices. In both settings we use multiparametric approach to define intermediate exponents. Then we split the weighted version of Dyson's transference theorem and an analogue of Khintchine's transference theorem for Diophantine exponents of lattices into chains of inequalities between the intermediate exponents we define b…
Families of Unit Equations and Exponential Diophantine Problems via Integral Points
本文通过Ru-Vojta定理和Huang-Levin-Xiao不等式推广研究了投影簇上整数点的分布，证明了两类一参数单位方程解集的退化性，并扩展了GCD估计以获得特定指数迪菲antine方程和q进制表示中数字分布的新结果。
Geometry of Diophantine exponents
本综述探讨了丢番图指数及其满足的关系，介绍了经典丢番图指数、带权与乘性类似物及格的指数，并重点关注转换原理。
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.
Let $\mathcal{C}$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation with two independent approximation functions; that is if a certain sum converges then the set of all points $(x,y)$ on the curve which satisfy simultaneously the inequalities $\| q x \| < ψ_1(q)$ and $\| qy \| < ψ_2(q)$ infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation $\| qx \| \| q y \| < ψ(q)$, we establish a Hausdorff measure convergence re…
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。
In this paper we adapt parametric geometry of numbers developed by Wolfgang Schmidt and Leonard Summerer to a multiplicative setting, and derive a chain of inequalities for the corresponding exponents which splits the transference inequality for Diophantine exponents of lattices in the same way Khintchine's transference inequalities for simultaneous approximation can be split.
In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bmθ\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bmθ)= 1/…
In this paper we develop some of the ideas belonging to W.Schmidt and L.Summerer to define intermediate Diophantine exponents and split Dyson's transference inequality into a chain of inequalities for intermediate exponents. This splitting generalizes the analogous result of M.Laurent and Y.Bugeaud for Khintchine's transference inequalities.
Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we show that German's transference inequalities between the two most classical exponents of uniform Diophantine approximation are optimal. Further, we establish that the spectrum of the $n$ uniform exponents of Diophantine approximation in dimension $n$ is a subset of $\mathbb{R}^n$ with non empty interior. Thus, no Jarník-type relation holds between them.
In this paper we aim to prove two inequalities involving the classical approximation constants $w_{n}^{\prime}(ζ),\hat{w}_{n}^{\prime}(ζ)$ that stem from the simultaneous approximation problem $|ζ^{j}x-y_{j}|$, $1\leq j\leq n$, on the one side and the constants $w_{n}^{\ast}(ζ),\hat{w}_{n}^{\ast}(ζ)$ connected to approximation with algebraic numbers of degree $\leq n$ on the other side. We concretely prove $w_{n}^{\ast}(ζ)\hat{w}_{n}^{\prime}(ζ)\geq 1$ and $\hat{w}_{n}^{\ast}(ζ)w_{n}^{\prime}(ζ)\geq 1$. The first result is due to W. Schmidt, however our method of proving it allows to derive th…
In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior that these functions exhibit and prove a Central Limit Theorem for them. Our approach is based on a quantitative study of higher-order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of Siegel transforms.
A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation
本文提出了一种从球体到开集的维度质量转移原理，用于解决动力学迪奥菲尼逼近中的缩小目标问题，并揭示了热力学正式主义与该领域的新联系。
The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the Jarník-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. Th…
Rational points close to non-singular algebraic curves
该研究聚焦于非奇异三元型对应的丢番图不等式解的密度问题，等价探讨接近非奇异平面代数曲线的有理点密度，为数论相关领域的有理点分布研究提供方向。
Effective Results in The Metric Theory of Quantitative Diophantine Approximation
本文将丢番图逼近度量理论中依赖隐式常数的渐近定量化为有效结果，在测度至多 δ 的例外集外显式控制常数，并推出 Schmidt 定理、Koukoulopoulos–Maynard 定理及正规数等结果。
We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $α$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in …
We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For $S$ consisting of more than one prime, finite or infinite, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprindžuk conjectures using nondivergence estimates for flows on homogeneous spaces i…
Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures
本文研究吕罗良逼近数的度量性质，补充证明谭-周测度论命题，用质量转移原理推广曹-吴-张维数结果，构造反例修正谭-周维数猜想并给出部分结果。
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