Diophantine Approximation Research Articles

An upper bound of the Hausdorff dimension of singular vectors on affine subspaces

In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920’s. We study the set of singular vectors on an affine subspace of R n \mathbb {R}^n . We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace.

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.

Diophantine approximation by rational numbers of certain parity types

AbstractFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

Fractal dimensions of the Markov and Lagrange spectra near $3$

The Lagrange spectrum \mathcal{L} and Markov spectrum \mathcal{M} are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. It is known that the Hausdorff dimensions of the intersections of these sets with any half-line coincide, that is, \dim_{H}(\mathcal{L}\cap (-\infty, t)) = \dim_{H}(\mathcal{M}\cap (-\infty, t))\equalscolon d(t) for every t \geq 0 . It is also known that d(3)=0 and d(3+\varepsilon)>0 for every \varepsilon>0 .We show that, for sufficiently small values of \varepsilon > 0 , one has the approximation d(3+\varepsilon) = 2\cdot\frac{W(e^{c_0}|\log \varepsilon|)}{|\log \varepsilon|}+\mathrm{O}(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^{2}}) , where W denotes the Lambert function (the inverse of f(x)=xe^{x} ) and c_{0}=-\log\log((3+\sqrt{5})/2) \approx 0.0383 . We also show that this result is optimal for the approximation of d(3+\varepsilon) by “reasonable” functions, in the sense that, if F(t) is a C^{2} function such that d(3+\varepsilon) = F(\varepsilon) + \mathrm{o}(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^{2}}) , then its second derivative F''(t) changes sign infinitely many times as t approaches 0 .

Uniform Diophantine approximation and run-length function in continued fractions

Abstract We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set $$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$ for a class of quadratic irrational numbers $y\in [0,1)$ . These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.

GCD inequalities inspired by Vojta’s conjecture

AbstractWe prove several GCD inequalities inspired by Vojta’s deep conjecture in Diophantine geometry. These inequalities work with all algebraic numbers, so they can be thought of as attempts to extend GCD inequalities for S-units obtained by Corvaja and Zannier. Some of the inequalities we derive are weakened versions of inequalities obtained by Silverman under the assumption of Vojta’s conjecture. The upper bounds for the GCD’s in these inequalities all involve some contributions of heights and some contributions coming from the local heights outside S. The main ingredient of the proofs is the recent Diophantine approximation result of Ru and Vojta which is based on Schmidt subspace theorem and which involves a birational invariant coming from the asymptotic growth of the number of global sections of certain line bundles.

Diophantine approximation with prime denominator in quadratic number fields under GRH

Diophantine approximation with prime denominator in quadratic number fields under GRH

Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer’s confluent hypergeometric functions.

On a theorem of Borel on diophantine approximation

On a theorem of Borel on diophantine approximation

A note on Diophantine approximation with four squares and one k-th power of primes

A note on Diophantine approximation with four squares and one k-th power of primes

On discrepancy, intrinsic Diophantine approximation, and spectral gaps

On discrepancy, intrinsic Diophantine approximation, and spectral gaps

Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers

Abstract Let (Fn ) n≥0 and (Ln ) n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10 d Fm + Lk and Fn = 10 d Lm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk , respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.

On the Connection Between Irrationality Measures and Polynomial Continued Fractions

AbstractLinear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence $${F}_{n}={F}_{n-1}+{F}_{n-2}$$ F n = F n - 1 + F n - 2 , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $$\zeta \left(3\right)$$ ζ 3 , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as $$\pi$$ π and $$e$$ e . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $$\pi$$ π , $$e$$ e , $$\zeta \left(3\right)$$ ζ 3 , and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $$\zeta \left(5\right)$$ ζ 5 ).

On Robustness for the Skolem, Positivity and Ultimate Positivity Problems

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

Lévy-Khintchin Theorem for best simultaneous Diophantine approximations

Lévy-Khintchin Theorem for best simultaneous Diophantine approximations

Littlewood and Duffin–Schaeffer-Type Problems in Diophantine Approximation

Gallagher’s theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully inhomogeneous version of Gallagher’s theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin–Schaeffer conjecture for a class of nonmonotonic approximation functions.

Uniform Diophantine approximation with restricted denominators

Let b≥2 be an integer and A=(an)n=1∞ be a strictly increasing subsequence of positive integers with η:=lim supn→∞an+1an<+∞. For each irrational real number ξ, we denote by vˆb,A(ξ) the supremum of the real numbers vˆ for which, for every sufficiently large integer N, the equation ‖banξ‖<(baN)−vˆ has a solution n with 1≤n≤N. For every vˆ∈[0,η], let Vˆb,A(vˆ) (Vˆb,A⁎(vˆ)) be the set of all real numbers ξ such that vˆb,A(ξ)≥vˆ (vˆb,A(ξ)=vˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ). When η=1, we prove that the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) are equal to (1−vˆ1+vˆ)2 for any vˆ∈[0,1]. When η>1 and limn→∞⁡an+1an exists, we show that the Hausdorfff dimension of Vˆb,A(vˆ) is strictly less than (η−vˆη+vˆ)2 for some vˆ, which is different with the case η=1, and we give a lower bound of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) for any vˆ∈[0,η]. Furthermore, we show that this lower bound can be reached for some vˆ.

Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation

The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that dim H ⁡ W ( ψ ) × ⋯ × W ( ψ ) = d − 1 + dim H ⁡ W ( ψ ) \begin{equation*} \dim _{\mathcal H}W(\psi )\times \cdots \times W(\psi )=d-1+\dim _{\mathcal H}W(\psi ) \end{equation*} where W ( ψ ) W(\psi ) is the set of ψ \psi -well approximable points in R \mathbb {R} and ψ : N → R + \psi : \mathbb {N}\to \mathbb {R}^+ is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand’s famous inequality on the dimension of the Cartesian product of general sets.

A note on weighted simultaneous Diophantine approximation on manifolds

A note on weighted simultaneous Diophantine approximation on manifolds