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

Roy, Damien

Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 211-224

Résumé

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 $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$ .

Détail

DOI : 10.4153/CJM-2007-009-3

Mots-clés : 11J13, 11J82

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