Metric Theory Of Diophantine Approximation Research Articles

Metric theory of diophantine approximation and asymptotic estimates for the number of polynomials with given discriminants divisible by a large power of a prime number

Discriminants of polynomials characterize the distribution of roots of polynomials in the complex plane. In recent years, for integer polynomials, exact lower-bound estimates have been obtained for the number of polynomials of a given degree and height. The method of obtaining these estimates is based on Minkowski’s theorems in the geometry of numbers and the metric theory of Diophantine approximation. A new method is proposed and allows one to obtain upperbound estimates for the number of polynomials with bounded discriminants in Archimedean and non-Archimedean metrics. The method generalizes the ideas of H. Davenport, B. Volkman, and V. Sprindzuk that allowed them to obtain significant advances in solving Mahler’s problem.

A Note on Well Distributed Sequences

Abstract We prove an easy statement about inhomogeneous approximation for non-singular vectors in metric theory of Diophantine approximation.

Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables

The article is devoted to the latest results in metric theory of Diophantine approximation. One of the first major result in area of number theory was a theorem by academician Jonas Kubilius. This paper is dedicated to centenary of his birth. Over the last 70 years, the area of Diophantine approximation yielded a number of significant results by great mathematicians, including Fields prize winners Alan Baker and Grigori Margulis. In 1964 academician of the Academy of Sciences of BSSR Vladimir Sprindžuk, who was a pupil of academician J. Kubilius, solved the well-known Mahler’s conjecture on the measure of the set of S-numbers under Mahler’s classification, thus becoming the founder of the Belarusian academic school of number theory in 1962.

Polynomials with small values in the neighborhoods of zeros in Archimedean and non-Archimedean metrics

For a positive integer 𝑄 > 0, let 𝐼 ⊂ R denote an interval of length 𝜇1𝐼 = 𝑄−𝛾1 (where 𝜇1 is the Lebesgue measure) and 𝜇2𝐾 = 𝑄−𝛾2 , 𝛾2 > 0 (where 𝜇2 is the Haar measure of a measurable cylinder 𝐾 ⊂ Q𝑝). Let us denote the set of polynomials of degree ≤ 𝑛 and height 𝐻 (𝑃) ≤ 𝑄 as 𝒫𝑛 (𝑄) = {𝑃 ∈ Z[𝑥] : deg 𝑃 ≥ 𝑛, 𝐻 (𝑃) ≤ 𝑄} . Let 𝒜(𝑛,𝑄) denote the set of real and 𝑝-adic roots of such polynomials 𝑃 (𝑥) lying in the space 𝑉 = 𝐼 ×𝐾. In this paper it is proved that the following inequality holds for a suitable constant 𝑐1 = 𝑐1 (𝑛) and 0 ≤ 𝑣1, 𝑣2 6 1 2 : #𝒜(𝑛,𝑄) > 𝑐1𝑄𝑛+1−𝛾1−𝛾2 . The proof relies on methods of metric theory of Diophantine approximation developed by V.G. Sprindzuk to prove Mahler’s conjecture and by V.I. Bernik to prove A. Baker’s conjecture.

Тригонометрические суммы в метрической теории диофантовых приближений

Это обзор результатов по метрической теории диофантовых приближений на многообразиях в n-мерном евклидовом пространстве, в доказательстве которых используются тригонометрические суммы.Мы приводим как классические теоремы, так и современные результаты для многообразий Γ, dim Γ = m, n/2 < m < n. Мы также показываем, как происходит переход от задачи о диофантовых приближениях к оценке тригонометрической суммы или тригонометрического интеграла, и приводим необходимые соображения теории меры.Если m ≤ n/2, то обычно используют другие методы. Например, метод существенных и несущественных областей или методы эргодической теории.Здесь даны две фундаментальные теоремы рассматриваемой теории. Одну из них в 1977 г. доказал В. Г. Спринджук. Другую теорему в1998 г. получили Д. И. Клейнбок и Г. А. Маргулис. Первая теорема была доказана методом тригонометрических сумм. Вторая теорема – методами эргодической теории. Для ее доказательства авторами была найдена связь между диофантовыми приближения и однородными динамическими системами.В заключении кратко упоминаем о тенденциях развития метрической теории диофантовых приближений зависимых величин, даем ссылки на ее современные аспекты.

Counting real algebraic numbers with bounded derivative of minimal polynomial

In this paper, we consider the problem of counting algebraic numbers [Formula: see text] of fixed degree [Formula: see text] and bounded height [Formula: see text] such that the derivative of the minimal polynomial [Formula: see text] of [Formula: see text] is bounded, [Formula: see text]. This problem has many applications to the problems of metric theory of Diophantine approximation. We prove that the number of [Formula: see text] defined above on the interval [Formula: see text] does not exceed [Formula: see text] for [Formula: see text] and [Formula: see text]. Our result is based on an improvement to a lemma from Gelfond’s monograph “Transcendental and algebraic numbers”. Given an integer polynomial small enough in some point, the lemma provides an upper bound for the absolute value of its irreducible divisor. We obtain a stronger estimate which holds in real points located far enough from all algebraic numbers of bounded degree and height. This is done by considering the resultant of two polynomials represented as the determinant of the Sylvester matrix for the shifted counterparts.

Trigonometric sums in the metric theory of Diophantine approximation

Trigonometric sums in the metric theory of Diophantine approximation

THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS

We consider sequences of the form $\left(a_{n} \alpha\right)_{n}$ mod 1, where $\alpha\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\alpha$ in the sense of Lebesgue measure, we say that $(a_n)_n$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_n)_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_n)_n$ having large additive energy which, however, maintains the metric pair correlation property.

ON THE ESTIMATES OF RESULTANT VALUES OF INTEGER POLYNOMIALS WITHOUT COMMON ROOTS

In the article we present an improvement to the lemma on the order of simultaneous zero approximation by the values of two integer polynomials without common roots from A. O. Gelfond’s monograph “Transcendental and algebraic numbers”. The lemma says that if two integer polynomials P1 and P2 of degree not exceeding n1 and n2 and of height not exceeding Qµ1 and Qµ2 respectively having no roots in common take values 1 Px Q 1( ) −τ < and 2 Px Q 2 ( ) −τ < at some transcendental point x ∈¡, then min( , ) τ τ < µ + µ +δ 1 2 12 21 n n . Gelfond’s lemma and similar results have important applications to many problems of the metric theory of Diophantine approximation. One of such applications is the result due to V. Bernik (1983) on the upper bound for the Hausdorff dimension of the set of real numbers with specified order of zero approximation by the values of integer polynomials. This result along with the result of A. Baker and W. Schmidt (1970) on the lower bound of the Hausdorff dimension of the set mentioned above gives the exact formula. In order to prove the upper bound V. Bernik improved and extended Gelfond’s lemma by considering the values of polynomials of degree not exceeding n and of height not exceeding Qµ on some interval of length Q−η and obtaining a stronger inequality τ+µ+ τ+µ−η < µ +δ 2max( , 0) 2 , n τ= τ τ min( , ). 1 2 However, the need to consider the same estimates for the degree and height of the polynomials is still restrictive and limits the range of problems this result could be applied to. In our work we consider the values of polynomials of different degrees and heights on an interval and obtain a stronger estimate by using higher order derivatives, thus improving and extending Gelfond’s lemma and existing similar results. The result is obtained using the methods of the theory of transcendental numbers.

An Inhomogeneous Jarník type theorem for planar curves

AbstractIn metric Diophantine approximation there are classically four main classes of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental to each of them. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximation on manifolds for each of the classes above. In particular, both Khintchine and Jarník-type results have been established for approximation on planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves in the setting of dual approximation and in so doing complete the metric theory of Diophantine approximation on planar curves.

On metric theory of Diophantine approximation for complex numbers

On metric theory of Diophantine approximation for complex numbers

Metrical Diophantine approximation for quaternions

AbstractAnalogues of the classical theorems of Khintchine, Jarník and Jarník-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general ‘lim sup’ sets.

Metric Diophantine approximation on homogeneous varieties

AbstractWe develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.

The Convergence Part of a Knintchine-Type Theorem in the Ring of Adeles

Abstract We prove the convergence part of a Khintchine-type theorem for simultaneous Diophantine approximation of zero by values of integral polynomials at the points (x, z, ω1, ω2) ∈ R × C × Qp1 × Qp2 , where p1 ≠ p2 are primes. It is a generalization of Sprindžuk’s problem (1980) in the ring of adeles. We continue our investigation (2013), where the problem was proved at the points in R2 × C × Qp1 . We use the most precise form of the essential and inessential domains method in metric theory of Diophantine approximation.

SIMULTANEOUS DIOPHANTINE APPROXIMATION IN R2 × C × Qp

ABSTRACT An analogue of the convergence part of Khintchine’s theorem (1924) for simultaneous approximation of integral polynomials at the points (x1, x2, z,w) ∈ R2 × C × Qp is proved. It is a solution of the more general problem than Sprindźuk's problem (1980) in the ring of adeles. We use a new form of the essential and nonessential domain methods in metric theory of Diophantine approximation

Simultaneous diophantine approximation in $\mathbb{R}^2 × \mathbb{C} × \mathbb{Q}_p$

An analogue of the convergence part of Khintchine's theorem (1924) for simultaneous approximation of integral polynomials at the points $(x_1,x_2,z,w)\in\mathbb{R}^2\times\mathbb{C}\times\mathbb{Q}_p$ is proved. It is a solution of the more general problem than Sprind\u{z}uk's problem (1980) in the ring of adeles. We use a new form of the essential and nonessential domains method in metric theory of Diophantine approximation.

METRICAL RESULTS ON SYSTEMS OF SMALL LINEAR FORMS

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine–Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximation function.

THE METRIC THEORY OF MIXED TYPE LINEAR FORMS

In this paper the metric theory of Diophantine approximation of linear forms that are of mixed type is investigated. Khintchine–Groshev theorems are established together with the Hausdorff measure generalizations. The latter includes the original dimension results obtained in [H. Dickinson, The Hausdorff dimension of sets arising in metric Diophantine approximation, Acta Arith.68(2) (1994) 133–140] as special cases.

METRIC CONSIDERATIONS CONCERNING THE MIXED LITTLEWOOD CONJECTURE

The main goal of this paper is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan–Teulié Conjecture — also known as the "Mixed Littlewood Conjecture". Let p be a prime. A consequence of our main result is that, for almost every real number α, [Formula: see text]

The distribution of close conjugate algebraic numbers

AbstractWe investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.