Tong's spectrum for Rosen continued fractions (Q933163)

Let \(\lambda= \lambda_q=2\cos{\pi\over q}\) with a positive integer \(q\geq 3\). For real \(x\), put \(r(x)= \lfloor{\varepsilon\over\lambda x}+{1\over 2}\rfloor\), \(\varepsilon= \text{sgn}(x)\), and define a map \(f_q\) of the interval \([-{\lambda\over 2},{\lambda\over 2})\) into itself by \(f_q(0)= 0\), \(f_q(x)={\varepsilon\over x}-\lambda r(x)\) for \(x\neq 0\). This yields the Rosen continued fraction expansion, or \(\lambda\)-expansion of \(x\), \[ x= {\varepsilon_1\over r_1\lambda+} {\varepsilon_2\over r_2\lambda+} {\varepsilon_3\over r_3\lambda+ \cdots}. \] It was introduced by \textit{D. Rosen} [Duke Math. J. 21, 549--563 (1954; Zbl 0056.30703)] in his study of the discontinuous groups which are generated by the transformations \(\tau\mapsto\tau+ \lambda_q\), \(\tau\mapsto -{1\over\tau}\) of the upper half plane. The case \(q= 3\) gives the nearest integer continued fraction expansion of \(x\) which was studied, among others, by \textit{J. C. Tong} [Math. Scand. 71, No. 2, 161--166 (1992; Zbl 0787.11028)] and ibid. 74, No. 1, 17--18 (1994; Zbl 0812.11042 )]. Let \({R_n\over S_n}= {\varepsilon_1\over r_1\lambda+} {\varepsilon_2\over r_2\lambda+} \cdots{\varepsilon_n\over r_n\lambda}\) be the convergents for \(x\) obtained from truncating the \(\lambda\)-expansion, where \(R_n\), \(S_n\) are real cyclotomic integers, and call \(\Theta_n= \Theta_n(x)= S^2_n|x-{R_n\over S_n}|\) the Rosen approximation coefficients of \(x\). A real number \(x\) is called \(G_q\)-rational if \(x={R_n\over S_n}\) for some \(n\), and \(G_q\)-irrational otherwise. \textit{A. Haas} and \textit{C. Series} [J. Lond. Math. Soc., II. Ser. 34, 219--234 (1986; Zbl 0605.10018)] proved a Hurwitz type result saying that for every \(G_q\)-irrational \(x\) there exist infinitely many \(G_q\)-rationals \({r\over s}\) such that \(s^2|x- {r\over s}|< H_q\), where \(H_q={1\over 2}\) if \(q\) is even and \(H_q= {1\over 2}(1+(1- {\lambda\over 2})^2)^{-1/2}\) if \(q\) is odd. In the paper under review the authors prove a Borel type result: For every \(G_q\)-irrational \(x\) there are infinitely many \(n\) for which \(\Theta_n\leq H_q\), and \(H_q\) is best possible. In the case \(q= 3\) of nearest integer continued fractions (when \(R_n\), \(S_n\) are rational integers), \textit{J. C. Tong} (loc. cit.) showed that for all real irrational \(x\) and all positive integers \(n\), \(k\) one has \[ \min\{\Theta_{n-1},\Theta_n,\dots, \Theta_{n+k}\}> c_k= {1\over\sqrt{5}}+ {1\over\sqrt{5}}\Biggl({3- \sqrt{5}\over 2}\Biggr)^{2k+3}, \] where the constant \(c_k\) is best possible. The authors prove similar results for all \(q\geq 3\). For even \(q= 2p\geq 4\) it says that for every \(G_q\)-irrational \(x\) and all \(n\), \(k\) one has \[ \min\{\Theta_{n-1}, \Theta_n,\dots, \Theta_{n+k(p-1)}\}< c_{q,k-1} \] with some explicitely specified constant \(c_{q,k-1}\) which is best possible. An essential part of the paper and of the proofs is occupied by a study of the dynamics of the map \(T_q(x, y)= (f_q(x),{1\over r\lambda+\varepsilon y})\) and of regions in \(\mathbb{R}^2\) where \(T_q\) is bijective almost everywhere.