Effective Irrationality Measures for Certain Algebraic Numbers

On irrationality measures of [formula omitted

Let \(P \in \mathbb {Z}[X,Y]\) be a square-free polynomial of total degree d with integer coefficients of bitsize less than τ and $$\displaystyle \mathrm {V}_{\mathbb {R}}(P):= \{ (\alpha , \gamma ) \in \mathbb {R}^2~|~P(\alpha , \gamma )=0 \} $$ be the real planar curve defined as the vanishing set of P. In this paper, assuming that \(\mathrm {V}_{\mathbb {R}}(P)\) contains no vertical line, we give two quantitative results. The first one is a lower bound on the minimum distance between two different roots of the polynomial P(X, γ + y), where y is any small enough real number. This length is called the separator of P(X, γ + y). The second result provides a lower bound on the distance between the most closer root of P(X, γ + y) to the vertical tangent at (α, γ), assuming that \(\mathrm {V}_{\mathbb {R}}(P)\) contains also no horizontal line. This length is called the deviation of the curve from the vertical tangent. The proofs of these results use recent quantitative results on algebraic numbers appearing in Diatta, et al. (Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology, 2018. arXiv:1807.10622), which also play a crucial role in the study of the complexity of computing the topology of \(\mathrm {V}_{\mathbb {R}}(P)\) in the approach of Diatta, et al. (Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology, 2018. arXiv:1807.10622).

We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel′dman and Rickert, proving, for example, that \[ \max \left \{ \left | \sqrt {2} - p_{1}/q \right | , \left | \sqrt {3} - p_{2}/q \right | \right \} > q^{-1.79155} \] for $q > q_{0}$ (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell’s equations and norm form equations.

A result of Chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given. More particularly, Padé approximants to the function ( 1 − x ) 1 / 3 {(1 - x)^{1/3}} are employed to show, for certain integers a and b , that \[ | ( a / b ) 1 / 3 − p / q | > c q − κ when q > 0. \left | {{{(a/b)}^{1/3}} - p/q} \right | > c{q^{ - \kappa }}\quad {\text {when}}\,q > 0. \] Here, c and k are given as functions of a and b only.

A vector m = ( m 1 , … , m n ) ∈ Z n ∖ { 0 } m = ({m_1}, \ldots ,{m_n}) \in {{\mathbf {Z}}^n}\backslash \{ 0\} is called an integer relation for the real numbers α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} , if ∑ α i m i = 0 \sum {\alpha _i}{m_i} = 0 holds. We present an algorithm that, when given algebraic numbers α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} and a parameter ε \varepsilon , either finds an integer relation for α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} or proves that no relation of Euclidean length shorter than 1 / ε 1/\varepsilon exists. Each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation. Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs \[ poly ( log ⁡ 1 / ε , n , log ⁡ max i height ( α i ) , [ Q ( α 1 , … , α n ) : Q ] ) {\operatorname {poly}}\left ( {\log 1/\varepsilon ,n,\log \max \limits _i {\text {height}}({\alpha _i}),[{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}):{\mathbf {Q}}]} \right ) \] bit operations. The straightforward algorithm that works with a primitive element of the field extension Q ( α 1 , … , α n ) {\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}) of Q would take \[ poly ( n , log ⁡ max i height ( α i ) , ∏ i = 1 n degree ( α i ) ) {\operatorname {poly}}\left ( {n,\log \max \limits _i {\text {height}}({\alpha _i}),\prod \limits _{i = 1}^n {{\text {degree}}({\alpha _i})} } \right ) \] bit operations. In order to prove the correctness of the algorithm, we show a lower bound for | ∑ α 1 m i | \left | {\sum {\alpha _1}{m_i}} \right | if m is not an integer relation for α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} , which may be interesting in its own right.

Diophantische Approximationen

In this paper, we present a result on using algebraic conjugates to form a sequence of approximations to an algebraic number, and in this way obtain effective irrationality measures for related algebraic numbers. From this result, we are able to generalise Thue's Fundamentaltheorem.

A vector m=(m1,...,m n ) ∈ Zn {0} is called an integer relation for the real numbers α1,...,α n , if Σα i m i =0 holds. We present an algorithm that when given algebraic numbers α1,...,α n and a parameter ɛ either finds an integer relation for α1,...,α n or proves that no relation of euclidean length shorter than 1/ɛ exists. Each algebraic number is assumed to be given by its minimal polynomial and by a rational approximation precise enough to separate it from its conjugates.

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

For a complex number x , ∥ x ∥ := min ⁡ { | x - m | : m ∈ ℤ } {\|x\|:=\min\{|x-m|:m\in\mathbb{Z}\}} . Let k ≥ 1 {k\geq 1} be an integer, and let K be a number field. Let α 1 , … , α k {\alpha_{1},\dots,\alpha_{k}} be algebraic numbers with | α i | ≥ 1 {|\alpha_{i}|\geq 1} and let d i {d_{i}} denotes the degree of α i {\alpha_{i}} for 1 ≤ i ≤ k {1\leq i\leq k} . Set d = d 1 + ⋯ + d k {d=d_{1}+\cdots+d_{k}} . In this article, we show that if the inequality 0 < ∥ λ 1 ⁢ q ⁢ α 1 n + ⋯ + λ k ⁢ q ⁢ α k n ∥ < θ n q d + ε {0<\|\lambda_{1}q\alpha^{n}_{1}+\cdots+\lambda_{k}q\alpha^{n}_{k}\|<\frac{% \theta^{n}}{q^{d+\varepsilon}}} has infinitely many solutions in ( n , q , λ 1 , … , λ k ) ∈ ℕ 2 × ( K × ) k {(n,q,\lambda_{1},\dots,\lambda_{k})\in\mathbb{N}^{2}\times(K^{\times})^{k}} with absolute logarithmic Weil height of λ i {\lambda_{i}} is small compared to n and some θ ∈ ( 0 , 1 ) {\theta\in(0,1)} , then, in particular, the tuple ( λ 1 ⁢ q ⁢ α 1 n , … , λ k ⁢ q ⁢ α k n ) {(\lambda_{1}q\alpha^{n}_{1},\dots,\lambda_{k}q\alpha^{n}_{k})} is pseudo-Pisot, and at least one of α i {\alpha_{i}} is an algebraic integer. This result can be viewed as Roth-type theorem for linear combinations of powers of algebraic numbers over ℚ ¯ {\overline{\mathbb{Q}}} . The case q = 1 {q=1} was recently proved in [A. Kulkarni, N. M. Mavraki and K. D. Nguyen, Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier, Trans. Amer. Math. Soc. 371 2019, 6, 3787–3804], which is a generalization of Mahler’s question proved in [P. Corvaja and U. Zannier, On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mendès France, Acta Math. 193 2004, 2, 175–191]. As a consequence of our result, we obtain the following generalization of this question: let α > 1 {\alpha>1} be an algebraic number with d = [ ℚ ( α ) : ℚ ] {d=[\mathbb{Q}(\alpha):\mathbb{Q}]} . For a given ε > 0 {\varepsilon>0} , if the inequality 0 < ∥ λ ⁢ q ⁢ α n ∥ < θ n q d + ε 0<\|\lambda q\alpha^{n}\|<\frac{\theta^{n}}{q^{d+\varepsilon}} has infinitely many solutions in the tuples ( n , q , λ ) ∈ ℕ 2 × K × {(n,q,\lambda)\in\mathbb{N}^{2}\times K^{\times}} with absolute logarithmic Weil height of λ is small compared to n and θ ∈ ( 0 , 1 ) {\theta\in(0,1)} , then some power of α is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers.

Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a family of double complex integrals, we prove ℚ(α)-irrationality measures and ℚ(α)-nonquadraticity measures of log α for several algebraic numbers α∈ℂ, improving earlier results due to Amoroso and the second-named author.

Algebraic numbers and irrationality measures

The uniform measure of simple regular sets of infinite trees

Let Q denote the rational number field and α be an algebraic number of degree s. Then the algebraic number field F s = Q(α) is the field given by the polynomials in α of degree < s with rational coefficients.

The Siegel conjecture on the rational approximation to algebraic numbers was proved a few years ago by K. F. Roth [1] with the following theorem: Let α be any algebraic number, not rational. If has an infinity of solutions in integers h and q (q &gt; 0) tehn k ≤ 2.