Effective results for restricted rational approximation to quadratic irrationals (Q2919686)

Let \(b\geq 2\) an integer, and \(\xi\) a real algebraic number of degree \(d \geq 2\). The authors study combinatorial properties of the \textit{\(b\)-ary expansion} of \(\xi\). Let \(\|x\|\) denotes the distance from a real number \(x\) to the nearest integer. \textit{A. Schinzel} [Acta Arith. 13, 177--236 (1967; Zbl 0159.07101)] proved the following:NEWLINENEWLINE({t1}) For every integer \(b\geq 2\) and every quadratic real number \(\xi\), we have \(\|b^n\xi\| > b^{-n} \exp\{c(\xi, b)n^{1/7}\}\) for \(n \geq 1\), where \(c(\xi, b)\) is a positive effectively computable constant depending only on \(\xi\) and \(b\).NEWLINENEWLINE Let \(v_b(\xi)\) the infimum of the real numbers \(v\) for which the inequality \(\|b^n\xi\| > (b^n)^{-v}\) holds for every sufficiently large positive integer \(n\). Let \(v_b^{\mathrm{eff}}(\xi)\) be the infimum of the real numbers \(v\) for which there exists an effectively computable constant \(c(\xi, v)\) such that \(\|b^n\xi\| > c(\xi, v)(b^n)^{-v}\) for \(n \geq 1\). Since the publication of Schinzel's paper (1967) [loc. cit.], there have been important improvements in the theory of linear forms in two \textit{\(p\)-adic logarithms} which allow the authors to improve theorem ({t1}):NEWLINENEWLINE({t2}). For every integer \(b\geq 2\) and every quadratic real number \(\xi\), we have \(\|b^n\xi\| > c(\xi, b)b^{-(1-\tau(\xi,\mathcal P))n}\) for \(n\geq 1\), where \(c(\xi, b)\) and \(\tau (\xi,\mathcal P)\) are positive effectively computable constants depending only on \(\xi\) and \(b\), and on \(\xi\) and on the set \(\mathcal P\) of prime factors of \(b\), respectively. In particular, we have \(v_b^{\mathrm{eff}}(\xi)\leq 1-\tau(\xi,\mathcal P)\).NEWLINENEWLINE The theory of linear forms in logarithms enables them to improve theorem ({t2}) to the following uniform result.NEWLINENEWLINE({t3}) Let \(p\) be a prime number. There exists an effectively computable absolute positive constant \(\tau_1\) such that \(v_p^{\mathrm{eff}}(\sqrt{p^2+1})\leq 1-\tau_1\).NEWLINENEWLINEFinally, the authors show that, for a special class of quadratic numbers, the exponent \(v_b^{\mathrm{eff}}\) can be very small, like in the following case:NEWLINENEWLINEFor every integers \(b\geq 2\) and \(k\geq 1\), we have \(v_b^{\mathrm{eff}}(\sqrt{b^{2k}+1})\leq (\log 48)/(k\log b)\).NEWLINENEWLINEIn the introduction the authors recall the story since Schinzel, in particular the contributions of \textit{N. I. Fel'dman} [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 973--990 (1971; Zbl 0237.10018), English transl.: Math. USSR, Izv. 5, 985--1002 (1972; Zbl 0259.10031)], the sharpening of \textit{A. Baker}'s theory of logarithms in linear forms [Mathematika 13, 204--216 (1966; Zbl 0161.05201); ibid. 14, 102--107 (1967; Zbl 0161.05202); ibid. 220--228 (1967; Zbl 0161.05301); ibid. 15, 204--216 (1968; Zbl 0169.37802)]. The results of \textit{M. Laurent} et al. [J. Number Theory 55, No. 2, 285--321 (1995; Zbl 0843.11036)] and \textit{E. Bombieri} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, No. 1, 61--89 (1993; Zbl 0774.11034)] are also recalled.