Metric Diophantine Approximation—From Continued Fractions to Fractals

Introduction During the last 15–20 years it has been realized that certain problems in Diophantine approximation and number theory can be solved using geometry of the space of lattices and methods from the theory of flows on homogeneous spaces. The purpose of this survey is to demonstrate this approach on several examples. We will start with Diophantine approximation on manifolds where we will briefly describe the proof of Baker–Sprindžuk conjectures and some Khintchine-type theorems. The next topic is the Oppenheim conjecture proved in the mid-1980s and the Littlewood conjecture, still not settled. After that we will go to quantitative generalizations of the Oppenheim conjecture and to counting lattice points on homogeneous varieties. In the last part we will discuss results on unipotent flows on homogeneous spaces which play, directly or indirectly, the most essential role in the solution of the above-mentioned problems. Most of those results on unipotent flows are proved using ergodic theorems and also notions such as minimal sets and invariant measures. These theorems and notions have no effective analogs and because of that the homogeneous space approach is not effective in a certain sense. We will briefly discuss the problem of the effectivization at the very end of the paper. The author would like to thank A. Eskin and D. Kleinbock for their comments on a preliminary version of this article. Diophantine approximation on manifolds We start by recalling some standard notation and terminology.

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.

In 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.

We 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.

We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems.

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.

We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

Metric Diophantine approximation over a local field of positive characteristic

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.

Analogues 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.

Preface. A problem of Steinhaus concerning the existence of a plane set with a certain property S.D. Adhikari. Self affine tiling and Pisot numeration system S. Akiyama. A fundamental but unexploited partition invariant K. Alladi. On Algebraic independence of certain functions related to the elliptic modular function M. Amou. Fragments by Ramanujan on Lambert Series B.C. Berndt. Metric theory of Diophantine approximation in the field of complex numbers V.I. Bernik, M.M. Dodson. The Davenport-Heilbronn Fourier transform method, and some diophantine inequalities J. Brudern. On the probabilistic complexity of numerically checking the binary Goldbach conjecture in certain intervals J.M. Deshouillers, H. te Riele. On the mean square of Hecke L-functions associated to holomorphic cusp forms S. Egami. Mean Square of an Error Term Related to a Certain Exponential Sum Involving the Divisor Function J. Furuya. On zeros of the Lerch zeta-function R. Garunkstis, A. Laurincikas. Power values of products of consecutive integers and binomial coefficients K. Gyory. A note on Hilbert modular threefolds Y. Hamahata. Inverse Galois Problem for Dihedral Groups K. Hashimoto, K. Miyake. On Ramachandra's method for the mean value problems of various L-functions Y. Kamiya. On the zeros of certain modular forms M. Kaneko. A weighted integral approach to the mean square of Dirichlet L-functions M. Katsurada, K. Matsumoto. The mean value theorem of the Riemann zeta-function in the critical strip for short intervals I. Kuchi, Y. Tanigawa. On inhomogeneous Diophantine approximation and the NST-algorithm T. Komatsu. Selberg zeta functions of PGL and PSL over function fields S. Koyama. A Survey on the Number Field Sieve K. Nakamula. Non-normal class number one problem and the least prime power-residue R. Okazaki. Higher dimensional modular equations of degree 7 R. Sasaki. Exponential congruences A. Schinzel. Pade approximation for words generated by certain substitutions, and Hankel determinants J. Tamura. On Sturmian Sequences which are invariant under some substitutions S. Yasutomi.

The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol’d-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical properties of these exceptional sets are closely related to fundamental results in the metrical theory of Diophantine approximation. The counterpart of Diophantine approximation in hyperbolic space and a dynamical interpretation which led to the very general notion of’ shrinking targets’ are sketched and the recent use of flows in homogeneous spaces of lattices in the proof of the Baker-Sprindzuk conjecture is described briefly.KeywordsHomogeneous SpaceHyperbolic SpaceHausdorff DimensionRotation NumberDiophantine ApproximationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequalityfor almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, andfis a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint onf, the number of solutions of (1) is asymptotically equal toasktends to infinity.

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]