Simultaneous Rational Approximation Research Articles

On certain simultaneous rational approximations in $${\textbf {R}}^2$$ and certain rational approximations in $${\textbf {Q}}_p$$

On certain simultaneous rational approximations in $${\textbf {R}}^2$$ and certain rational approximations in $${\textbf {Q}}_p$$

Padé approximation for a class of hypergeometric functions and parametric geometry of numbers

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We explicitly construct Padé approximations by using a formal method and show that the associated sequences satisfy a Poincaré-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an effective version of the Poincaré-Perron theorem. As a consequence we obtain, among others, effective irrationality measures for values of binomial functions at rational numbers, which might have useful arithmetic applications. A general theorem on simultaneous rational approximations that we need is proven by using new arguments relying on parametric geometry of numbers.

Simultaneous rational approximation to successive powers of a real number

We develop new tools leading, for each integer n ≥ 4 n\ge 4 , to a significantly improved upper bound for the uniform exponent of rational approximation λ ^ n ( ξ ) \widehat {\lambda }_n(\xi ) to successive powers 1 , ξ , … , ξ n 1,\xi ,\dots ,\xi ^n of a given real transcendental number ξ \xi . As an application, we obtain a refined lower bound for the exponent of approximation to ξ \xi by algebraic integers of degree at most n + 1 n+1 . The new lower bound is n / 2 + a n + 4 / 3 n/2+a\sqrt {n}+4/3 with a = ( 1 − log ⁡ ( 2 ) ) / 2 ≃ 0.153 a=(1-\log (2))/2\simeq 0.153 , instead of the current n / 2 + O ( 1 ) n/2+\mathcal {O}(1) .

Periodic representations for quadratic irrationals in the field of 𝑝-adic numbers

Continued fractions have been widely studied in the field of p p -adic numbers Q p \mathbb Q_p , but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard, for any quadratic irrational via p p -adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R \mathbb R and Q p \mathbb Q_p . Moreover given two primes p 1 p_1 and p 2 p_2 , using the Binomial transform, we are also able to pass from approximations in Q p 1 \mathbb {Q}_{p_1} to approximations in Q p 2 \mathbb {Q}_{p_2} for a given quadratic irrational. Then, we focus on a specific p p –adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.

On simultaneous rational approximation to a p-adic number and its integral powers, II

AbstractLet $p$ be a prime number. For a positive integer $n$ and a real number $\xi$, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

Effective simultaneous rational approximation to pairs of real quadratic numbers

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|, \|q \zeta\| \} > q^{-1 + \tau}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.

A transference principle for simultaneous rational approximation

We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important inequality of Marnat and Moshchevitin which describes the spectrum of the pairs of ordinary and uniform exponents of rational approximation to those points. For points whose pair of exponents are close to the boundary in the sense that they almost realize the equality, we provide additional information about the corresponding sequence of best rational approximations. We conclude with an application.

A class of maximally singular sets for rational approximation

We say that a subset of [Formula: see text] is maximally singular if its contains points with [Formula: see text]-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to [Formula: see text], the maximal possible value. In this paper, we give a criterion which provides many such sets including Grassmannians. We also recover a result of the author and Roy about a class of quadratic hypersurfaces.

On Infinitely Many Rational Approximants to ζ(3)

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

An equivalence principle between polynomial and simultaneous Diophantine approximation

We show that Mahler's classification of real numbers $\zeta$ with respect to the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(\lambda_{n}(\zeta))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximation and metric theory. We deduce estimates on the Hausdorff dimension of well-approximable vectors on the Veronese curve and refine the best known upper bound for the exponent $\widehat{\lambda}_{n}(\zeta)$ for even $n\geq 4$.

On the simultaneous rational approximations to real and p-adic numbers

We consider simultaneous rational approximations to real and p-adic numbers. We prove that for any irrational number α0 and p-adic number α, there are infinitely many irreducible fractions satisfying ∣α0 − m/n ∣ 0 is an arbitrary number, and ∣ ⋅ ∣ p is the p-adic norm.

A new exponent of simultaneous rational approximation

We introduce a new exponent of simultaneous rational approximation $\widehat{\lambda}_{\min}(\xi,\eta)$ for pairs of real numbers $\xi,\eta$, in complement to the classical exponents $\lambda(\xi,\eta)$ of best approximation, and $\widehat{\lambda}(\xi,\eta)$ of uniform approximation. It generalizes Fischler's exponent $\beta_0(\xi)$ in the sense that $\widehat{\lambda}_{\min}(\xi,\xi^2) = 1/\beta_0(\xi)$ whenever $\lambda(\xi,\xi^2) = 1$. Using parametric geometry of numbers, we provide a complete description of the set of values taken by $(\lambda,\widehat{\lambda}_{\min})$ at pairs $(\xi,\eta)$ with $1$, $\xi$, $\eta$ linearly independent over $\mathbf{Q}$.

Continued fractions and Parametric geometry of numbers

Recently, W. M. Schmidt and L. Summerer developed a new theory called Parametric Geometry of Numbers which approximates the behaviour of the successive minima of a family of convex bodies in ℝ n related to the problem of simultaneous rational approximation to given real numbers. In the case of one number, we show that the qualitative behaviour of the minima reflects the continued fraction expansion of the smallest distance from this number to an integer.

Simultaneous rational approximation via Rickert’s integrals

Using Rickert’s contour integrals, we give effective lower bounds for simultaneous rational approximations to numbers in the sets{(1−aN)ν,(1+aN)ν}and{(1+aN)ν,(1+2aN)ν}. Here N>a≥1 are integers, 0<ν<1 is a rational number and at least one of the radicals is irrational in each set. The result is valid for all q≥1 where q denotes the denominator of the approximating rational number.

Recurrent properties of quasi-periodic dynamical systems with multiple frequencies of p-adic Liouville numbers

We introduce a class of p-adic Liouville numbers by using Schneider’s continued fractions with the orders, which specify the goodness of rational approximations. Considering the simultaneous rational approximations of the p-adic number and its powers, we give some inequality relations between these orders and the exponents of Diophantine approximations. As typical examples, given by using these inequality relations, we treat quasi-periodic dynamical systems, the frequencies of which are these p-adic numbers, and we investigate recurrent properties of their orbits. We obtain the positive gap values between the upper and the lower recurrent dimensions, which show some unpredictability properties of the orbits. Furthermore, we give symbolic dynamical descriptions for these properties.

Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to d real numbers α1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter t as t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in t.

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.

RATIONAL APPROXIMATIONS TO VALUES OF BELL POLYNOMIALS AT POINTS INVOLVING EULER’S CONSTANT AND ZETA VALUES

AbstractIn this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

On inhomogeneous diophantine approximation and Hausdorff dimension

Let Γ = Z A + Z n ⊂ R n be a dense subgroup of rank n + 1 and let $$ \hat{w} $$ (A) denote the exponent of uniform simultaneous rational approximation to the generating point A. For any real number v ≥ $$ \hat{w} $$ (A), the Hausdorff dimension of the set B v of points in R n that are v-approximable with respect to Γ is shown to be equal to 1/v.

On simultaneous rational approximation to ap-adic number and its integral powers

AbstractLetpbe a prime number. For a positive integernand ap-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integersqsuch that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖pare all less thanq−λ−1. Here, ‖x‖pdenotes the infimum of |x−n|pasnruns through the integers. We study the set of values taken by the function λn.