×

Effective simultaneous rational approximation to pairs of real quadratic numbers. (English) Zbl 1455.11095

The author extends the usual concept of effective irrationality measures to simultaneous approximation to pairs of real numbers by rational numbers with the same denominator. Let \(\xi,\zeta\) be real numbers such that \(1,\xi,\zeta\) are linearly independent over \(\mathbb Q\). \(\mu\) is a simultaneous effective irrationality measure for the pair \((\xi,\zeta)\), if there exists an effectively computable \(c(\xi,\zeta)\geq 0\) such that, for every integer triple \((p,q,r)\) with \(q\ge 1\), we have \[\max\left\{\left|\xi-\frac{p}{q}\right|,\left|\zeta-\frac{p}{q}\right|\right\}\geq \frac{c(\xi,\zeta)}{q^{\mu}}.\] Denote by \(\mu_{\mathrm{eff}}(\xi,\zeta)\) the infimum of the effective irrationality measures for the pair \((\xi,\zeta)\) and call it the effective irrationality exponent of the pair \((\xi,\zeta)\).
The author proves the following statement:
Let \(\xi,\zeta\) be real quadratic numbers in distinct quadratic fields. Let \(R_{\xi}\) and \(R_{\zeta}\) denote the regulators of the fields \(\mathbb Q(\xi)\) and \(\mathbb Q(\zeta)\), respectively. There exists an effectively computable real number \(c_1\geq 0\) such that \[\mu_{\mathrm{eff}}(\xi,\zeta)\le 2- (c_1R_{\xi}R_{\zeta})^{-1}.\] In particular, if \(a,b\) are positive integers, none of \(a,b,ab\) is a perfect square, then there exists an effectively computable real number \(c_2\geq 0\) such that \[\mu_{\mathrm{eff}}(\sqrt{a},\sqrt{b})\le 2- (c_2\sqrt{ab}(\log a)(\log b))^{-1}.\] The proof applies Baker type lower estimates for linear forms in the logarithms of algebraic numbers.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11D09 Quadratic and bilinear Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.1093/qmath/15.1.375 · Zbl 0222.10036 · doi:10.1093/qmath/15.1.375
[2] 10.1093/qmath/20.1.129 · Zbl 0177.06802 · doi:10.1093/qmath/20.1.129
[3] ; Bennett, Number theory. CMS Conf. Proc., 15, 55 (1995)
[4] 10.1090/S0002-9947-96-01480-8 · Zbl 0873.11042 · doi:10.1090/S0002-9947-96-01480-8
[5] 10.5802/jtnb.262 · Zbl 1010.11036 · doi:10.5802/jtnb.262
[6] ; Bombieri, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20, 61 (1993) · Zbl 0774.11034
[7] 10.1017/CBO9780511542961.014 · doi:10.1017/CBO9780511542961.014
[8] 10.1006/jnth.1998.2245 · Zbl 0926.11015 · doi:10.1006/jnth.1998.2245
[9] 10.4171/183 · Zbl 1394.11001 · doi:10.4171/183
[10] ; Bugeaud, Osaka J. Math., 55, 315 (2018) · Zbl 1441.11015
[11] 10.1112/S0025579317000298 · Zbl 1434.11042 · doi:10.1112/S0025579317000298
[12] 10.4064/aa170828-7-3 · Zbl 1410.11025 · doi:10.4064/aa170828-7-3
[13] ; Feldman, Mat. Sb. (N.S.), 77, 423 (1968)
[14] ; Feldman, Izv. Akad. Nauk SSSR Ser. Mat., 35, 973 (1971) · Zbl 0237.10018
[15] 10.1017/S0305004100064598 · Zbl 0641.10014 · doi:10.1017/S0305004100064598
[16] 10.1017/S0305004100076118 · Zbl 0786.11040 · doi:10.1017/S0305004100076118
[17] 10.1007/BF02392078 · Zbl 0173.04801 · doi:10.1007/BF02392078
[18] 10.4153/CJM-1993-010-1 · Zbl 0774.11036 · doi:10.4153/CJM-1993-010-1
[19] 10.1007/978-3-662-11569-5 · doi:10.1007/978-3-662-11569-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.