Theory Of Diophantine Approximation Research Articles

Accumulation points of normalized approximations

Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points qα with α∈Rd and q∈Z. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of α whose accumulation points are all of Rd. In the second part we focus primarily on the case when the coordinates of α together with 1 form a basis for an algebraic number field K. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when d=2, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in K) of a single ellipse, or of a pair of hyperbolas, depending on whether or not K has a non-trivial embedding into C.

Weighted twisted inhomogeneous diophantine approximation

Abstract We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let ∑ i = 1 m α i = m and | ⋅ | α = max 1 ⩽ i ⩽ m | ⋅ | 1 / α i . Given an n-tuple of monotonically decreasing functions Ψ = ( ψ 1 , … , ψ n ) with ψ i : R + → R + such that each ψ i ( r ) → 0 as r → ∞ and fixed A ∈ R n × m define W A ( Ψ ) := { b ∈ [ 0 , 1 ] n : | A i ⋅ q − b i − p i | < ψ i ( | q | α ) ( 1 ⩽ i ⩽ n ) , for infinitely many ( p , q ) ∈ Z n × ( Z m ∖ { 0 } ) } . We prove that the set W A ( Ψ ) has zero-full Lebesgue measure under convergent–divergent sum conditions with some mild assumptions on A and the approximating functions Ψ. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are the weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (2023 Adv. Math. 428 109154), and Wang & Wu (2021 Math. Ann. 381 243–317) respectively.

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.

Различие мер Хаара цилиндров в теореме Дирихле для поля p-адических чисел

The Dirichlet box principle gives surprisingly accurate results in problems of approximation of real numbers by rational numbers, transcendental numbers by real algebraic numbers. Every polynomial taking small values at a given point $x$ also takes small values in its neighborhood. A problem of studying such neighborhoods and obtaining possible Lebesgue measure values arises frequently. In this paper we solve the problem in the p-adic case using recent results of the metric theory of Diophantine approximations.

Tiling of rectangles with squares and related problems via Diophantine approximation

This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using the theory of Diophantine Approximation, hence overcoming long-established difficulties, such as generalizations to higher-dimensional analogues. The universality of the method is demonstrated through its applications to different tiling problems. These include tiling rectangles with other rectangles, with their respective higher-dimensional counterparts, as well as tiling equilateral triangles, parallelograms, and trapezoids with equilateral triangles.

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

Upper bounds and spectrum for approximation exponents for subspaces ofℝn

This paper uses W. M. Schmidt’s idea formulated in 1967 to generalise the classical theory of Diophantine approximation to subspaces of ℝ n . Given two subspaces of ℝ n , A and B of respective dimensions d and e, with d+e⩽n, the proximity between A and B is measured by t=min(d,e) canonical angles 0⩽θ 1 ⩽⋯⩽θ t ⩽π/2; we set ψ j (A,B)=sinθ j . If B is a rational subspace, its complexity is measured by its height H(B)=covol(B∩ℤ n ). We denote by μ n (A|e) j the exponent of approximation defined as the upper bound (possibly equal to +∞) of the set of β>0 such that for infinitely many rational subspaces B of dimension e, the inequality ψ j (A,B)⩽H(B) -β holds. We are interested in the minimal value μ ˚ n (d|e) j taken by μ n (A|e) j when A ranges through the set of subspaces of dimension d of ℝ n such that for all rational subspaces B of dimension e one has dim(A∩B)<j. We show that if A is included in a rational subspace F of dimension k, its exponent in ℝ n is the same as its exponent in ℝ k via a rational isomorphism F→ℝ k . This allows us to deduce new upper bounds for μ ˚ n (d|e) j . We also study the values taken by μ n (A|e) e when A is a subspace of ℝ n satisfying dim(A∩B)<e for all rational subspaces B of dimension e.

On the metric theory of multiplicative Diophantine approximation

In 1962, Gallagher proved a higher-dimensional version of Khintchine’s theorem on Diophantine approximation. Gallagher’s theorem states that for any non-increasing approximation function ψ: ℕ → (0, 1/2) with $$\sum\nolimits_{q = 1}^\infty {\psi \left( q \right)} $$ and γ = γ′ = 0 the following set $$\left\{ {\left( {x,y} \right) \in {{\left[ {0,\,1} \right]}^2}:\left\| {qx - \gamma } \right\|\left\| {qy - {\gamma ^\prime }} \right\| < \psi \left( q \right)\,\,{\rm{infinitely}}\,{\rm{often}}} \right\}$$ has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on γ, γ′) of the above result. In this paper, we prove an Erdős—Vaaler type result for fibred multiplicative Diophantine approximation. Along the way, via a different method, we prove a slightly weaker version of Chow—Technau’s theorem with the condition that at least one of γ, γ′ is not Liouville. We also extend Chow—Technau’s result for fibred inhomogeneous Gallagher’s theorem for Liouville fibres.

Factorizable subgroups of the circle group

A subgroup H of the circle group T is said to be a-characterized if there exists a strictly increasing sequence of positive integers (un)n∈N0, with un|un+1 for all n∈N0, such that H consists precisely of those elements x∈T with unx→0 in T. These subgroups appeared in the study of trigonometric series in harmonic analysis, as well as in Diophantine approximation, dynamical systems and ergodic theory. The aim of the paper is to show that any a-characterized subgroup of T can be presented as the sum of two of its proper a-characterized subgroups.

On a problem of Hardy and Littlewood in the theory of Diophantine approximations

On a problem of Hardy and Littlewood in the theory of Diophantine approximations

DETERMINACY OF SCHMIDT’S GAME AND OTHER INTERSECTION GAMES

AbstractSchmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$ , which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$ . One of our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha ,\beta ,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha , \beta , \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$ .

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.

Intrinsic Diophantine approximation on quadric hypersurfaces

We consider the question of how well points in a quadric hypersurface {M\subseteq\mathbb{R}^d} can be approximated by rational points of {\mathbb{Q}^d\cap M} . This contrasts with the more common setup of approximating points in a manifold by all rational points in {\mathbb{Q}^d} . We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani (1986) and Kleinbock–Margulis (1999).

How smooth is quantum complexity?

The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed.

Quantitative subspace theorem and general form of second main theorem for higher degree polynomials

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions.

Diophantine approximation with the constant right-hand side of inequalities on short intervals

In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | &lt; Q-w, w &gt; n, Q &gt;1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w &gt; n +1 we get the estimate µ B&lt; c1(n) Q – (w-1/n). The best estimate to date was c2(n) Q –(w- n/n).

Hamiltonian Floer Theory for Nonlinear Schrödinger Equations and the Small Divisor Problem

Abstract We prove the existence of infinitely many time-periodic solutions of nonlinear Schrödinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov–Floer compactness theorem to infinite dimensions, we show how to solve the arising small divisor problem by combining elliptic methods with results from the theory of diophantine approximations.

On exremality of affine image of topological product of some varieties

In the theory of Diophantine Approximations one considers a question on approximation of Real Numbers by rational fractions with one and the same denominators. Among intensively studied questions of this theory a special place occupy Metric questions. Here such questions of the theory are considered which take place for almost all real numbers from given interval. For the first time similar questions have been studied by Khintchine for approximation of independent quantities. It had been investigated by him conditions at which for almost all real numbers specified accuracy of approximation is reached. Very important in the technical plan Khintchince’s transference principle allows us to connect a simultaneous approximations of dependent quantities with approximations of linear forms with integral coefficients. In 1932 Mahler K. has entered classification of transcendental numbers into consideration. He showed that almost all transcendental numbers are 𝑆 -numbers. Moreover, Mahler had proved an existence of a constant 𝛾 > 0 such that for almost all 𝜔 $$|𝑃(𝜔)| > ℎ^(−𝑛𝛾)$$, for all polynomials 𝑃 of degree no more 𝑛 and height ℎ > ℎ0(𝜔, 𝑛, 𝛾). Mahler showed that it is possible to take $$𝛾 = 4 + 𝜀$$. In the same work Mahler made an assumption that it is possible to take 𝛾 = 1 + 𝜀 for almost all real numbers. This hypothesis was proved by Sprindzhuk V. G by a method of essential and nonessential domains. Simultaneously, Sprindzhuk V. G. advanced some hypotheses generalising and improving Mahler’s results. Further these investigations of Sprindzuk led to the development of a new direction in the theory of Diophantine Approximations – to the research of extremality of manifolds. In the present article we develop a new approach to these questions and offer a new proof for extremality of algebraic varieties. This method allows to establish extremality of affine image of topological product of some varieties. Considering one particular case, we prove that the extremality for these varieties is possible to deduct from theorems on the convergence exponent of a special integral of Terry’s problem, using E.I. Kovalevskaya’s lemma. Further, we derive from the getting result particular case of the Sprindzuk’s hypothesis on extremality of varieties, induced by monomials of a polynomial in two variables.

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.