Effective simultaneous rational approximation to pairs of real quadratic numbers

TL;DR

This paper proves the existence of effective bounds for simultaneous approximation of pairs of quadratic real numbers from distinct quadratic fields, showing how closely they can be approximated by rationals.

Contribution

It establishes effective, computable bounds for simultaneous rational approximation to pairs of quadratic numbers in different quadratic fields.

Findings

Existence of positive real numbers  and c with effective bounds.

For all sufficiently large q, max of distances  and  exceeds q^{-1 + au}.

Results are explicitly computable and apply to pairs in distinct quadratic fields.

Abstract

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.

Equations

$$\max \{\Vert q\xi \Vert ,\Vert q\zeta \Vert \}>q^{-1+\tau },$$

$$\Bigl{|}\xi-{p\over q}\Bigr{|}>{c(\xi)\over q^{\mu}}.$$

$$\max\Bigl{\{}\Bigl{|}\xi-{p\over q}\Bigr{|},\Bigl{|}\zeta-{r\over q}\Bigr{|}\Bigr{\}}>{c(\xi,\zeta)\over q^{\mu}}.$$

$${\mu }_{\mathrm{eff}}(\xi ,\zeta )\le \max \{{\mu }_{\mathrm{eff}}(\xi ),{\mu }_{\mathrm{eff}}(\zeta )\}.$$

$$\max\Bigl{\{}\Bigl{|}\sqrt{2}-{p\over q}\Bigr{|},\Bigl{|}\sqrt{3}-{r\over q}\Bigr{|}\Bigr{\}}>{10^{-7}\over q^{1.913}},\quad\hbox{for integers $p,q,r\geq 1$},$$

$${\mu }_{\mathrm{eff}}(\xi ,\zeta )\le 2-(c_1{R}_{\xi }{R}_{\zeta }{)}^{-1}.$$

$${\mu }_{\mathrm{eff}}(\sqrt{a},\sqrt{b})\le 2-(c_2\sqrt{ab}(\log a)(\log b){)}^{-1}.$$

$$R_D<\sqrt{D}(1+\log \sqrt{D}),$$

$${\mu }_{\mathrm{eff}}(\xi )\le d-(c_3{R}_{\xi }\log R_{\xi }{)}^{-1}$$

$${\mu }_{\mathrm{eff}}(\xi )\le d-(c_4{R}_{\xi }{)}^{-1},$$

$$x^2-a{y}^2=u,z^2-b{y}^2=v$$

$$\max \{x,y,z\}\le (\max \{|u|,|v|,2\}{)}^{c_5\sqrt{ab}(\log a)(\log b)}.$$

$$\log A_{j}\geq\max\Bigl{\{}h(\alpha_{j}),{{\rm e}\over D}|\log\alpha_{j}|,{1\over D}\Bigr{\}},\qquad 1\leq j\leq n.$$

$$B^{\prime}\geq 3D,\quad B^{\prime}\geq\max_{1\leq j\leq n-1}\Bigl{\{}{|b_{n}|\over\log A_{j}}+{|b_{j}|\over\log A_{n}}\Bigr{\}}.$$

$$\eqalign{\log|b_{1}\log\alpha_{1}&+\cdots+b_{n}\log\alpha_{n}|\cr&\geq-2^{n+26}\,n^{3n+9}\,D^{n+2}\,\log(3D)\,\log A_{1}\ldots\log A_{n}\log B^{\prime}.\cr}$$

$$B=\max \{3D,|b_1|,\dots ,|b_n|\}$$

$$0<|1-A\xi |<{\mathrm{e}}^{-\kappa h(A\xi )}<1.$$

$$C=4.10^{19}{d}^4\frac{(\log 3d{)}^7}{\kappa }{\log }^{\ast }\frac{d}{\kappa },Q=(2tC{)}^t\prod _{i=1}^{t}h({\xi }_i),$$

$$h(\xi )\le 10Q\max \{h(A),Q\}.$$

$$x^2-a{y}^2=u,z^2-b{y}^2=v,in\; positive\; integersx,y,z.$$

$$U=\max \{|u|,|v|,2\}andX=\max \{x,y,z\}.$$

$$\alpha \ge |{\alpha }^{\sigma }|,\beta \ge |{\beta }^{\sigma }|,\alpha {\epsilon }^{-1}\le |{\alpha }^{\sigma }|\epsilon ,\beta {\eta }^{-1}\le |{\beta }^{\sigma }|\eta ,$$

$$x+y\sqrt{a}=\alpha {\epsilon }^m,\mathrm{and}z+y\sqrt{b}=\beta {\eta }^n,$$

$$2y\sqrt{a}=\alpha {\epsilon }^m-{\alpha }^{\sigma }{\epsilon }^{-m}$$

$$2y\sqrt{b}=\beta {\eta }^n-{\beta }^{\sigma }{\eta }^{-n}.$$

$$\Lambda =|\alpha {\beta }^{-1}\sqrt{b/a}{\epsilon }^m{\eta }^{-n}-1|=|{\alpha }^{\sigma }{\beta }^{-1}\sqrt{b/a}{\epsilon }^{-m}{\eta }^{-n}-{\beta }^{\sigma }{\beta }^{-1}{\eta }^{-2n}|.$$

$$U_0=\max \{U,ab,{\epsilon }^2,{\eta }^2\}$$

$${\alpha }^2\le |u|{\epsilon }^2\le U_0^2,{\beta }^2\le |v|{\eta }^2\le U_0^2,$$

$$\eqalign{h(\alpha\beta^{-1}\sqrt{b/a})&\leq h(\alpha)+h(\beta)+h(\sqrt{a})+h(\sqrt{b})\cr&\leq\log\alpha+\log\beta+(\log a)/2+(\log b)/2\leq 3\log U_{0}.\cr}$$

$$\max \{m\log \epsilon ,n\log \eta \}\ge 12\log U_0.$$

Full text

[TABLE]

where $\|\cdot\|$ denotes the distance to the nearest integer.

1. Introduction and results

Let $\xi$ be an irrational real number. The real number $\mu$ is an irrationality measure for $\xi$ if there exists a positive real number $c(\xi)$ such that every rational number ${p\over q}$ with $q\geq 1$ satisfies

[TABLE]

If, moreover, the constant $c(\xi)$ is effectively computable, then $\mu$ is an effective irrationality measure for $\xi$. We denote by $\mu(\xi)$ (resp., ${\mu_{\rm eff}}(\xi)$) the infimum of the irrationality measures (*resp., *effective irrationality measures) for $\xi$ and call it the irrationality exponent (*resp., *effective irrationality exponent) of $\xi$. It follows from the theory of continued fractions that $\mu(\xi)\geq 2$ and an easy covering argument shows that equality holds for almost all $\xi$, with respect to the Lebesgue measure. Furthermore, if $\xi$ is real algebraic of degree $d\geq 2$, then Liouville’s inequality implies that ${\mu_{\rm eff}}(\xi)\leq d$, while Roth’s theorem asserts that $\mu(\xi)=2$. To get better upper bounds for the effective irrationality exponents of algebraic numbers is a notorious challenging problem.

The first result of this type was obtained in 1964 by Alan Baker [??], who established that ${\mu_{\rm eff}}(\root 3 \of{2})\leq 2.955$, but his method applies only to a very restricted class of algebraic numbers. A few years later, in 1971, Feldman [??], by means of a refinement of the lower bounds for linear forms in logarithms of algebraic numbers established by Baker, proved that the effective irrationality exponent of an arbitrary real algebraic number of degree greater than two is strictly less than its degree; see also [??] for a proof depending on lower bounds for linear forms in only two logarithms. Subsequently, Bombieri [??, ??] gave in 1993 an alternative proof of Feldman’s result, completely independent of the theory of linear forms in logarithms and based on the Thue–Siegel Principle. Further results and bibliographic references can be found in [??], see in particular Section 4.10.

In this note, we are concerned with the simultaneous approximation to pairs of real numbers by rational numbers having the same denominator. We extend the above definition of (effective) irrationality exponent as follows. Let $\xi,\zeta$ be real numbers such that $1,\xi,\zeta$ are linearly independent over the rational numbers. The real number $\mu$ is a simultaneous irrationality measure for the pair $(\xi,\zeta)$ if there exists a positive real number $c(\xi,\zeta)$ such that, for every integer triple $(p,q,r)$ with $q\geq 1$, we have

[TABLE]

If, moreover, the constant $c(\xi,\zeta)$ is effectively computable, then $\mu$ is an effective irrationality measure for the pair $(\xi,\zeta)$. We denote by $\mu(\xi,\zeta)$ (resp., ${\mu_{\rm eff}}(\xi,\zeta)$) the infimum of the irrationality measures (*resp., *effective irrationality measures) for the pair $(\xi,\zeta)$ and call it the irrationality exponent (*resp., *effective irrationality exponent) of the pair $(\xi,\zeta)$.

Let $\xi,\zeta$ be real numbers such that $1,\xi,\zeta$ are linearly independent over the rational numbers. An easy application of Minkowski’s theorem implies that $\mu(\xi,\zeta)\geq{3\over 2}$ and a covering lemma shows that equality holds for almost all pairs $(\xi,\zeta)$, with respect to the planar Lebesgue measure. Schmidt [??] established that $\mu(\xi,\zeta)={3\over 2}$ if $\xi$ and $\zeta$ are both real and algebraic. His result is ineffective and gives no better information on ${\mu_{\rm eff}}(\xi,\zeta)$ than the obvious inequality

[TABLE]

The particular case where $\xi$ and $\zeta$ are quadratic numbers in distinct number fields is of special interest. The obvious upper bound ${\mu_{\rm eff}}(\xi,\zeta)\leq 2$ has been improved in some cases, in particular by Rickert [??] (see his paper for earlier references), who established among other results that

[TABLE]

and subsequently by Bennett [??, ??]. The method used in these papers applies only to a very restricted class of pairs $(\xi,\zeta)$ of quadratic numbers.

The purpose of the present note is to show how the theory of linear forms in logarithms (or, alternatively, Bombieri’s method) allows us to improve the trivial upper bound ${\mu_{\rm eff}}(\xi,\zeta)\leq 2$ for all quadratic real numbers $\xi$ and $\zeta$ in distinct quadratic fields.

[TABLE]

In particular, if $a,b$ are positive integers such that none of $a,b$, and $ab$ is a perfect square, then there exists an absolute, positive, effectively computable real number $c_{2}$ such that

[TABLE]

The last assertion of Theorem 1.1 is an immediate consequence of the first one, since for any square-free integer $D\geq 2$ the regulator $R_{D}$ of the quadratic field generated by $\sqrt{D}$ satisfies

[TABLE]

see e.g. [??].

Theorem 1.1 is by no means surprising. It is ultimately a consequence of the quantity $B^{\prime}$, which has its origin in Feldman’s papers [??, ??] and is the key tool for his effective improvement of Liouville’s bound; see Theorem 2.1 and the discussion below it. Other consequences of the quantity $B^{\prime}$ can be found in [??] and in the recent papers [??, ??, ??].

We present a proof of Theorem 1.1 together with a proof of a slightly weaker version of it, with $R_{\xi}R_{\zeta}$ replaced by $R_{\xi}R_{\zeta}\log(R_{\xi}R_{\zeta})$ in (1.1). For the latter result, we apply an estimate for linear forms in three logarithms, while the former is derived from a result of Bombieri [??] (and can also be derived from an estimate for linear forms in only two logarithms). This is in accordance with the improvements on Liouville’s bound obtained by these two methods. Namely, for an algebraic number $\xi$ of degree $d$ at least equal to $3$, denoting by $R_{\xi}$ the regulator of the number field generated by $\xi$, it follows from the theory of linear forms in logarithms and from Bombieri’s method, respectively, that there exist effectively computable, positive real numbers $c_{3}$ and $c_{4}$ such that

[TABLE]

and

[TABLE]

respectively; see e.g. [??].

The last assertion of Theorem 1.1 is equivalent to the following statement on systems of Pellian equations.

[TABLE]

satisfy

[TABLE]

2. Auxiliary results

As usual, $h(\alpha)$ denotes the (logarithmic) Weil height of the algebraic number $\alpha$. Our auxiliary result for the proof of (a slightly weaker version of) Theorems 1.1 and 1.2 is a particular case of Theorem 2.1 of [??], which essentially reproduces a theorem of Waldschmidt [??, ??].

[TABLE]

Let $B^{\prime}$ be a real number satisfying

[TABLE]

If $b_{1}\log\alpha_{1}+\cdots+b_{n}\log\alpha_{n}$ is nonzero, then we have

[TABLE]

The quantity $B^{\prime}$ in Theorem 2.1, which replaces the quantity

[TABLE]

occurring in earlier estimates of Baker, originates in Feldman’s papers [??, ??]. It is a consequence of the use of the functions $x\mapsto{x\choose k}$ instead of $x\mapsto x^{k}$ in the construction of the auxiliary function. The key point is the presence of the factor $\log A_{n}$ in the denominator in the definition of $B^{\prime}$. It is of great interest when $b_{n}=1$ and $\log A_{n}$ is large, since it then allows us, roughly speaking, to replace $B$ by $B/(\log A_{n})$.

The auxiliary result for the proof of Theorems 1.1 and 1.2 is a particular case of Theorem 2 of Bombieri [??]. Actually, since the dependence in the parameters $d$ and $\kappa$ occurring in this theorem has been improved in [??], we choose to quote below a particular case of Théorème 1 of [??].

[TABLE]

Setting

[TABLE]

we have the upper bound

[TABLE]

Bombieri’s original proof of Theorem 2.2 (upto the dependence on $d$ and $\kappa$) is independent of the theory of linear forms in logarithms. An alternative proof, given in [??], depends on lower estimates for linear forms in two logarithms (a careful reader can observe that, while the proof of Théorème 1 of [??] rests on estimates for linear forms in three logarithms, estimates for linear forms in two logarithms are enough to establish Theorem 2.2 above, and even with a better numerical constant, since we have assumed that ${\bf{K}}$ is a real number field) combined with a lemma of geometry of numbers from [??]. To deduce Theorem 2.2 from estimates for linear forms in two logarithms, the crucial ingredient is ultimately the presence of the factor $B^{\prime}$ in these estimates.

3. Proofs

We start with the proof of (a slightly weaker version of) Theorem 1.2. Let $a,b$ be positive integers such that $1,\sqrt{a},\sqrt{b}$ are linearly independent over the rationals. Let $u,v$ be nonzero integers and consider the system of Pellian equations

[TABLE]

Set

[TABLE]

It is well-known [??, ??] that the theory of linear forms in logarithms allows us to bound effectively $X$ in terms of $U$. Our goal is to show that we can get a bound which is polynomial in $U$.

Let ${\varepsilon}$ and $\eta$ be the fundamental totally positive units of the rings of integers of the fields ${\bf{Q}}(\sqrt{a})$ and ${\bf{Q}}(\sqrt{b})$, respectively, normalized to be greater than $1$. We note that $\xi$ and $\eta$ are at least equal to $(1+\sqrt{5})/2$.

Let $x$, $y$, and $z$ be positive integers satisfying (3.1). Since the norm over ${\bf{Q}}$ of $x+y\sqrt{a}$ (resp., $z+y\sqrt{b}$) is $u$ (resp., $v$), there exist nonnegative integers $m,n$ and algebraic numbers $\alpha$ in ${\bf{Q}}(\sqrt{a})$ and $\beta$ in ${\bf{Q}}(\sqrt{b})$ such that

[TABLE]

[TABLE]

where the superscript ${\cdot}^{\sigma}$ denotes the Galois conjugacy.

Since ${\varepsilon}^{\sigma}={\varepsilon}^{-1}$ and $\eta^{\sigma}=\eta^{-1}$, we have

[TABLE]

and

[TABLE]

Set

[TABLE]

Clearly, $\Lambda$ is nonzero.

Set

[TABLE]

Observe that $\alpha=|u|/|\alpha^{\sigma}|$, $\beta=|v|/|\beta^{\sigma}|$, (3.2), and (3.4) imply that

[TABLE]

and

[TABLE]

Assume first that

[TABLE]

Observe that (3.3), (3.4), and (3.5) imply that

[TABLE]

and

[TABLE]

thus, by (3.6), we get

[TABLE]

It then follows from Theorem 2.1 applied with $\alpha_{1}={\varepsilon}$, $\alpha_{2}=\eta$, $\alpha_{3}=\alpha\beta^{-1}\sqrt{b/a}$ that

[TABLE]

where we write $\log^{*}$ for the function $\max\{1,\log\}$. Here and below, the numerical constant implied by $\ll$ is positive, absolute, and effectively computable.

The combination of (3.9) with (3.10) gives

[TABLE]

We deduce that

[TABLE]

while $X\ll\log U_{0}$ if (3.6) is not satisfied.

Consequently, no matter if (3.6) holds or not, there exist an effectively computable positive real number $C_{1}$, depending only on $a$ and $b$, and an effectively computable positive, absolute real number $c_{6}$ such that

[TABLE]

Combined with the upper bound (1.2), this gives Theorem 1.2 upto an extra logarithmic factor.

For the proof of (a slightly weaker version of) Theorem 1.1, without any loss of generality, we may assume that $\xi,\eta$ are positive integers $a,b$ as above. Then, keeping our notation, it follows from (3.11) that there exists an effectively computable positive real number $C_{2}$, depending only on $a$ and $b$, such that

[TABLE]

Combined with (1.2), this completes the proof of Theorem 1.1 upto an extra logarithmic factor.

It remains for us to explain how to deduce Theorems 1.1 and 1.2 from Theorem 2.2, applied with $\Gamma$ being the subgroup generated by ${\varepsilon}$ and $\eta$,

[TABLE]

Note that

[TABLE]

Assume that (3.6) holds. By combining (3.6), (3.7), (3.8), and (3.12) we get

[TABLE]

It then follows from Theorem 2.2 that

[TABLE]

Since $h(A)\leq 3\log U_{0}$ and

[TABLE]

there exist an effectively computable positive real number $C_{3}$, depending only on $a$ and $b$, and an effectively computable positive, absolute real number $c_{7}$ such that

[TABLE]

By increasing $c_{7}$ and $C_{3}$ if necessary, we see that (3.13) also holds if (3.6) is not satisfied. Then, proceeding as below (3.11), we establish Theorems 1.1 and 1.2.

**Acknowledgements. ** The idea of this note came immediately at the end of the workshop co-organized by Andrej Dujella in Dubrovnik at the end of June 2019. I am very pleased to thank him and the speakers of Friday morning.

References

[1] A. Baker, Rational approximations to $\root 3 \of{2}$ and other algebraic numbers, Quart. J. Math. Oxford Ser. 15 (1964), 375–383.

[2] A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137.

[3] M. A. Bennett, Simultaneous approximation to pairs of algebraic numbers. In: Number theory (Halifax, NS, 1994), 55–65, CMS Conf. Proc., 15, Amer. Math. Soc., Providence, RI, 1995.

[4] M. A. Bennett, Simultaneous rational approximation to binomial functions, Trans. Amer. Math. Soc. 348 (1996), 1717–1738.

[5] Yu. Bilu et Y. Bugeaud, Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes, J. Théor. Nombres Bordeaux 12 (2000), 13–23.

[6] E. Bombieri, Effective Diophantine approximation on ${\bf G}_{m}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20 (1993), 61–89.

[7] E. Bombieri, Forty years of effective results in Diophantine theory. In: A panorama of number theory or the view from Baker’s garden (Zürich, 1999), 194–213, Cambridge Univ. Press, Cambridge, 2002.

[8] Y. Bugeaud, Bornes effectives pour les solutions des équations en $S$-unités et des équations de Thue-Mahler, J. Number Theory 71 (1998), 227–244.

[9] Y. Bugeaud, On the digital representation of integers with bounded prime factors, Osaka J. Math. 55 (2018), 315–324.

[10] Y. Bugeaud, Linear forms in logarithms and applications. IRMA Lectures in Mathematics and Theoretical Physics, 28. European Mathematical Society (EMS), Zürich, 2018.

[11] Y. Bugeaud and J.-H. Evertse, $S$-parts of terms of integer linear recurrence sequences, Mathematika 63 (2017), 840–851.

[12] Y. Bugeaud, J.-H. Evertse, and K. Győry, $S$-parts of values of of univariate polynomials, binary forms and decomposable forms at integral points, Acta Arith. 184 (2018), 151–185.

[13] N. I.  Fel’dman, Improved estimate for a linear form of the logarithms of algebraic numbers, Mat. Sb. 77 (1968), 393–406 (in Russian); English translation in Math. USSR. Sb. 6 (1968) 423–436.

[14] N. I.  Fel’dman, An effective refinement of the exponent in Liouville’s theorem, Iz. Akad. Nauk SSSR, Ser. Mat. 35 (1971), 973–990 (in Russian); English translation in Math. USSR. Izv. 5 (1971) 985–1002.

[15] M. J. Jacobson, Jr., R. F. Lukes, and H. C. Williams, An investigation of bounds for the regulator of quadratic fields, Experiment. Math. 4 (1995), 211–225.

[16] R. G. E. Pinch, Simultaneous Pellian equations, Math. Proc. Cambridge Philos. Soc. 103 (1988), 35–46.

[17] J. H. Rickert, Simultaneous rational approximations and related Diophantine equations, Math. Proc. Cambridge Philos. Soc. 113 (1993), 461–472.

[18] W. M. Schmidt, On simultaneous approximations of two algebraic numbers by rationals, Acta Math. 119 (1967), 27–50.

[19] M. Waldschmidt, Minorations de combinaisons linéaires de logarithmes de nombres algébriques, Canadian J. Math. 45 (1993), 176–224.

[20] M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables, Grundlehren Math. Wiss. 326, Springer, Berlin, 2000.

Yann Bugeaud

Institut de Recherche Mathématique Avancée, U.M.R. 7501

Université de Strasbourg et C.N.R.S.

7, rue René Descartes

67084 STRASBOURG   (France)

e-mail : bugeaud@math.unistra.fr