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On Two Exponents of Approximation Related to a Real Number and Its Square

Published online by Cambridge University Press:  20 November 2018

Damien Roy
Damien Roy*
Affiliation:
Département de Mathématiques, Université d'Ottawa, 585 King Edward, Ottawa, ON, K1N 6N5 e-mail: droy@uottawa.ca
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Abstract

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For each real number $\xi$, let $\widehat{{{\lambda }_{2}}}\left( \xi \right)$ denote the supremum of all real numbers $\text{ }\!\!\lambda\!\!\text{ }$ such that, for each sufficiently large $X$ , the inequalities $\left| {{x}_{0}} \right|\,\le \,X,\,\left| {{x}_{0}}\xi \,-\,{{x}_{1}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ and $\left| {{x}_{0}}{{\xi }^{2}}\,-\,{{x}_{2}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero, and let $\widehat{{{\omega }_{2}}}\left( \xi \right)$ denote the supremum of all real numbers $\omega $ such that, for each sufficiently large $X$, the dual inequalities $\left| {{x}_{0}}\,+\,{{x}_{1}}\xi \,+\,{{x}_{2}}{{\xi }^{2}} \right|\,\le \,{{X}^{-\omega }}$, $\left| {{x}_{1}} \right|\,\le \,X$ and $\left| {{x}_{2}} \right|\,\le \,X$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponents $\widehat{{{\lambda }_{2}}}\left( \xi \right)$ where $\xi$ ranges through all real numbers with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$ form a dense subset of the interval $\left[ 1/2,\,\left( \sqrt{5}\,-\,1 \right)/2 \right]$ while, for the same values of $\xi$, the dual exponents $\widehat{{{\omega }_{2}}}\left( \xi \right)$ form a dense subset of $\left[ 2,\,\left( \sqrt{5}\,+\,3 \right)/2 \right]$. Part of the proof rests on a result of V. Jarník showing that $\widehat{{{\lambda }_{2}}}\left( \xi \right)=1-{{\hat{\omega }}_{2}}{{\left( \xi \right)}^{-1}}$ for any real number $\xi$ with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$.

Keywords

11J1311J82
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 1 , 01 February 2007 , pp. 211 - 224
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Allouche, J.-P., Davison, J. L., Queffélec, M., Zamboni, L. Q., Transcendence of Sturmian or morphic continued fractions. J. Number Theory 91(2001), no. 1, 39–66.Google Scholar
[2] Arbour, B. and Roy, D., A Gel’fond type criterion in degree two. Acta Arith. 11(2004), no. 1, 97–103.Google Scholar
[3] Bugeaud, Y. and Laurent, M., Exponents of Diophantine approximation and sturmian continued fractions. Ann Inst. Fourier (Grenoble) 55(2005), no. 3, 773–804.Google Scholar
[4] Davenport, H. and Schmidt, W. M., Approximation to real numbers by quadratic irrationals. Acta Arith. 13(1967), 169–176.Google Scholar
[5] Davenport, H. and Schmidt, W. M., Approximation to real numbers by algebraic integers. Acta Arith. 15(1969), 393–416.Google Scholar
[6] Fischler, S., Spectres pour l’approximation d’un nombre réel et de son carré. C. R. Acad. Sci. Paris 339(2004), no. 10, 679–682.Google Scholar
[7] Jarník, V., Zum Khintchineschen Übertragungssatz. Trudy Tbilisskogo mathematicheskogo instituta im. A. M. Razmadze = Travaux de l’Institut mathématique de Tbilissi 3(1938), 193–212.Google Scholar
[8] Roy, D., Approximation simultanée d’un nombre et de son carré. C. R. Acad. Sci., Paris 336(2003), no. 1, 1–6.Google Scholar
[9] Roy, D., Approximation to real numbers by cubic algebraic integers. I. Proc. London Math. Soc. 88(2004), no. 1, 42–62.Google Scholar
[10] Roy, D., Approximation to real numbers by cubic algebraic integers. II. Ann. of Math. 158(2003), no. 3, 1081–1087.Google Scholar
[11] Roy, D., Diophantine approximation in small degree. In: Number Theory, CRM Proceedings and Lecture Notes 36, American Mathematical Society, Providence, RI, 2004, pp. 269–285.Google Scholar
[12] Schmidt, W. M., Diophantine Approximation, Lecture Notes in Mathematics 785, Springer-Verlag, Berlin, 1980.Google Scholar