Rational approximations to algebraic numbers

Diophantische Approximationen

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.

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.

Diophantine approximation is concerned with the quantitative study of the density of the rational numbers inside the real numbers. The Diophantine properties of a real number can be quantified through its approximation properties by rational (and more generally algebraic) numbers. For rational approximation, continued fractions provide an important tool in studying such properties. For higher dimensional problems and for algebraic approximation, different methods are needed. The metric theory of Diophantine approximation is concerned with the size of sets of numbers enjoying specified Diophantine properties. It is a general feature of the theory that most natural properties give rise to zero–one laws: the set of numbers enjoying the property in question is either null or full with respect to the Lebesgue measure. A more refined study of the null sets can be done using the notions of Hausdorff measure and dimension. Over the years, considerable work has gone into studying metric Diophantine approximation on subsets of $$\mathbb {R}^n$$ . The initial focus was on curves, surfaces and manifolds, but in recent years much effort has also gone into the study of fractal subsets. Already in the setting of rational approximation of real numbers, many problems which seem simple enough remain open. For instance, it is not known whether the Cantor middle third set contains an algebraic, irrational number (it is conjectured not to do so). In these notes, starting from the classical setup, I will work towards the study of metric Diophantine approximation on fractal sets. Along the way, we will touch upon some major open problems in Diophantine approximation, such as the Littlewood conjecture and the Duffin–Schaeffer conjecture; and newer methods originating in ergodic theory and dynamical systems will also be discussed. The required elements from fractal geometry will be covered.

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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 &gt; 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q &gt; 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 &gt; 2. Nothing further in this direction however has hitherto been proved.‡

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 \[ | ( a / b ) 1 / 3 − p / q | &gt; c q − κ when q &gt; 0. \left | {{{(a/b)}^{1/3}} - p/q} \right | &gt; c{q^{ - \kappa }}\quad {\text {when}}\,q &gt; 0. \] Here, c and k are given as functions of a and b only.

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 &lt; ∥ λ 1 ⁢ q ⁢ α 1 n + ⋯ + λ k ⁢ q ⁢ α k n ∥ &lt; θ n q d + ε {0&lt;\|\lambda_{1}q\alpha^{n}_{1}+\cdots+\lambda_{k}q\alpha^{n}_{k}\|&lt;\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 &gt; 1 {\alpha&gt;1} be an algebraic number with d = [ ℚ ( α ) : ℚ ] {d=[\mathbb{Q}(\alpha):\mathbb{Q}]} . For a given ε &gt; 0 {\varepsilon&gt;0} , if the inequality 0 &lt; ∥ λ ⁢ q ⁢ α n ∥ &lt; θ n q d + ε 0&lt;\|\lambda q\alpha^{n}\|&lt;\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.

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 ${(1 - x)^{1/3}}$ are employed to show, for certain integers a and b, that \[ \left | {{{(a/b)}^{1/3}} - p/q} \right | > c{q^{ - \kappa }}\quad {\text {when}} q > 0.\] Here, c and k are given as functions of a and b only.

A vector m=(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.

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and e =x0 denotes the fundamental solution ofx2−Dy2=−1, thenx2+1=Dy4 is solvable if and only ify0=A2, where A is an integer. Moreover, if e>64, thenx2+1=Dy4 has at most one positive integral solution (x, y).

The aim of these lectures is to give an account of results obtained from the application of Thue’s idea of comparing two rational approximations to algebraic numbers in order to show that algebraic numbers cannot be approximated too well by rational numbers. In particular we will give special attention to the problem of obtaining effective measures of irrationality, or types, for various classes of algebraic numbers.KeywordsAlgebraic NumberProduct FormulaHodge StructureGeneral HeightFundamental InequalityThese 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.

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequalityhas only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.