On finite special continued fractions (Q2862857)

The author discusses finite continued fractions -- regular or of special kinds -- and proves two theorems with many auxiliary lemmas. Preliminary notations: Let \(\Gamma _{r}\) with \(0<r<1/2\) be a lattice on the plane: NEWLINE\[NEWLINE\Gamma_{r}=\{(n-r.m, m)\mid n,m\in \mathbb{Z}\}.NEWLINE\]NEWLINE Let \(\Omega \) be a closed convex and bounded domain in the plane with piecewise smooth boundary, which contains a neighborhood of 0 and be symmetric with respect to the coordinate axes, i.e. NEWLINE\[NEWLINE(x_{1},x_{2})\in \Omega \Rightarrow (-x_{1}, x_{2}),(x_{1}, -x_{2}),(-x_{1}, -x_{2})\in\Omega.NEWLINE\]NEWLINE Let \(\mathfrak{T}\) be an affine transformation: NEWLINE\[NEWLINE(x_1,x_2)\rightarrow(t_1x_1, t_2x_2)=\mathfrak{T}(x_1,x_2)NEWLINE\]NEWLINE with positive numbers \(t_1\) and \(t_2\). Let \(\mathfrak{T}(\Omega)\) denote the set of points \(\{\mathfrak{T}(x_1,x_2)\mid (x_1,x_2\in\Omega\}\). The non-zero node \(\gamma = (\gamma_1, \gamma_2)\) of the lattice \(\Gamma_{r}\) is called minimum with respect to \(\Omega\), if for some transformation \(\mathfrak{T}\) we have:NEWLINENEWLINE 1) only nodes \(\gamma\) and \(-\gamma\) lie on the border of \(\mathfrak{T} (\Omega)\);NEWLINENEWLINE 2) in \(\mathfrak{T} (\Omega)\) there are no non-zero nodes of \(\Gamma_{r}\).NEWLINENEWLINE NEWLINEThe set of all such minima is denoted by \(\mathfrak{M}(\Gamma_{r},\Omega)\). Let \(\Omega_{\theta}\) be the set NEWLINE\[NEWLINE\left\{(x_1,x_2)\in\mathbb R^2\mid |x_1|^{\theta}+|x_2|^{\theta}\leq 1\right\}NEWLINE\]NEWLINE with \(\theta\in [1,\infty)\), studied by \textit{H. Minkowski} [Ann. Sci. Éc. Norm. Supér. (3) 13, 41--60 (1896; JFM 27.0170.01)]. She proves the following: NEWLINENEWLINETheorem 1. Let \(\psi (x, y)=0\) be a function on the border of \(\Omega\), for which there does not exist a transformation \(\mathfrak{T}\) with the property \(\Omega_{\infty}= \mathfrak{T} (\Omega)\). Denote by \((a_0,b_0),(a_1,b_1)\) points under the condition: \(\psi(2a_0,0) = \psi(a_0,b_0) = \psi(a_1,b_1) = \psi(0,2b_1)=0\). For all \(\alpha\) of \([0, 1]\) we observe the function \(\beta_{\Omega} = \beta(\alpha)\) with following properties: NEWLINE\[NEWLINE\begin{cases}\psi(u, v) = 0,&\text{if}\,\, \alpha_{0}\leq u\leq a_{1};\\ \psi(s, t) = 0, u =s \beta, t = \nu \alpha &\text{ if}\,\, a_{1}\leq s \leq 2a_{0};\\ \psi(x,y) = 0, x = s-u, y = t+v, &\text{if}\,\, 0 \leq x \leq a_{0}\end{cases}.NEWLINE\]NEWLINE Define functions \(g(\alpha)\) and \(\overline{g}(\beta)\) by equalities \(g(\alpha) = \frac{\beta(\alpha)}{1+\alpha\beta(\alpha)}\) and \(\overline{g}(\beta)=\frac{\alpha(\beta)}{1+\beta \alpha(\beta)}\), where \(\alpha=\alpha(\beta)\) is a function, inverse to \(\beta=\beta(\alpha)\). Let:NEWLINENEWLINE1) \(g(\alpha)\), \(\overline{g}(\beta)\) be continuously differentiable on the segments \([0, 1]\) and \([1/2, 1]\), respectively;NEWLINENEWLINE2) There do not exist numbers \(\alpha_1,\alpha_2\in [0,1]\), \(\beta_1,\beta_2\in [1/2,1]\) for which \(g(\alpha)=c_1+c_2\) \((c_1\neq 0)\) for all \(\alpha \in (\alpha_1, \alpha_2])\), \(\overline{g}(\beta) =\overline{c}_1\beta+\overline{c}_2\) \((\overline{c}_1\neq 0)\) for all \(\beta\in (\beta_1, \beta_2]\). Then for a natural number \(d>2\) we have the asymptotic formula NEWLINE\[NEWLINE\sum_{_{\substack{ a=1, \\ \gcd(a,d) = 1}}}^d \sharp\, \mathfrak{M}(\Gamma_{a/d};\Omega) = \varphi(d)(\phi_{1}(\Omega)\log d)\;+(\phi_{2}(\Omega))+O_{\Omega,\varepsilon}(d^{5/6}\log^{7/6+\varepsilon}d),NEWLINE\]NEWLINE where \(\varepsilon\) is an arbitrary small positive number;NEWLINENEWLINE\(\phi_{1}(\Omega)=\frac{4\Phi (\Omega)}{\zeta (2)}\) (\(\phi_{1}(\Omega)>0\)) and \(\phi_{2}(\Omega)=\frac{4\Phi(\Omega)}{\zeta(2)}(2\gamma -2\frac{\zeta'(2)}{\zeta(2)}-1) \frac{4C_{1}(\Omega)}{\zeta(2)}+6\) where NEWLINE\[NEWLINE\Phi (\Omega)= \iint_{\overline{\Omega}}\frac{d\alpha d\beta}{(1+\alpha \beta)^{2}},NEWLINE\]NEWLINE \(\zeta\) is the zeta-function of Riemann, \(\gamma\) is the Euler constant, NEWLINE\[NEWLINEC_{1}(\Omega)=h(\Omega)-\frac{\log^{2}2}{2}-\int_\frac{1}{2}^1\frac{\log(1+\beta \alpha (\beta))}{\beta (1+\beta \alpha(\beta))}\,d\beta,NEWLINE\]NEWLINE \(h(\Omega)=h-h_{\alpha}(\Omega)+h_{\beta}(\Omega)\), \(h=\sum_{n=1}^{\infty}\frac{1}{n}(\sum_{n\leq m<2n}\frac{1}{n}-\log 2)\), NEWLINE\[NEWLINEh_{\alpha}(\Omega)=\sum_{n=1}^{\infty}\frac{1}{n}\left(\sum_{n/2\leq m\leq n} \frac{\alpha(m/n)}{n+m\alpha(m/n)}-\log 2+\Phi(\Omega)\right),NEWLINE\]NEWLINE NEWLINE\[NEWLINEh_{\beta}(\Omega)=\sum_{n=1}^{\infty}\frac{1}{n}\left(\sum_{m\leq 2n}\frac{\beta(m/n)}{n+m\beta(m/n)}-\Phi(\Omega)\right).NEWLINE\]NEWLINE The author studies continued fractions of the form \(b_{0}+\frac{a_{1}|}{|b_{1}}+\frac{a_{2}|}{|b_{2}}+\ldots+\frac{a_{s}|}{|b_{s}}\) with \(a_i\in \{-1,1\}\), \(b_0\in \mathbb Z\), \(b_i\in\mathbb N\) for all \(i\geq 1\), \(b_s\geq 2\) and its convergents NEWLINE\[NEWLINE\frac{P_{i}}{Q_{i}}=b_0+\frac{a_1|}{|b_1}+\frac{a_2|}{|b_2}+\ldots+\frac{a_{i-1}|}{|b_{i-1}}NEWLINE\]NEWLINE for \(i\geq 1\), \(P_0=1\), \(Q_0=0\). This continued fraction is called generalized \(\Omega\)-fraction of the number \(r\) \((0<r<1/2)\), if the finite \(\{a_{i}\}\) and \(\{b_{i}\}\) have the following two properties: 1) \(b_0=0\); 2) for all \(P_i-rQ_i\neq 0\) for \(i\geq 1\) the numbers \(a_i\), \(b_i\) are such that the node \(\gamma _{\Omega}^{(i+1)}=a_{i}\gamma_{\Omega}^{(i-1)}+b_{i}\gamma _{\Omega}^{(i)}\) is adjacent to \(\gamma _{\Omega}^{(i)}\) in \(\mathfrak{M}(\Gamma_{r}, \Omega)\) (i.e. there is a transformation \(\mathfrak{T}\), for which: 1) \(\pm\gamma _{\Omega}^{(i+1)}\) and \(\pm \gamma _{\Omega}^{(i)}\) lie on the border of \(\mathfrak{T}(\Omega)\) and 2) in the \(\mathfrak{T}(\Omega)\) there are no non-zero nodes of \(\Gamma_{r}\)). The fraction \(\frac{1|}{|1}+\frac{a_{1}|}{|b_{1}}+\frac{a_{2}|}{|b_{2}}+\ldots+\frac{a_{s}|}{|b_{s}}\) is called generalized \(\Omega\)-fraction of the number \(r\) \((1/2<r<1)\), if \(\frac{a_{1}}{|b_{1}+1}+\frac{a_{2}|}{|b_{2}}+\ldots+\frac{a_{s}|}{|b_{s}}\) is a generalized \(\Omega\)-fraction of the number \(1-r\). The relation between \(s = s(r,\Omega)\) is: \(s(r,\Omega) = \sharp\,\mathfrak{M}(\Gamma_{r};\Omega)/2-2\) if \(r<1/2\) and \(s(r,\Omega) = \sharp\,\mathfrak{M}(\Gamma_{1-r};\Omega)/2-1\) if \(r>1/2\). NEWLINENEWLINETheorem 2. Under the conditions in Theorem 1 we have NEWLINE\[NEWLINE\sum_{_{\substack{ a=1,\\ \gcd(a,d)=1}}}^d s(a/d,\Omega)=\varphi(d)\left(\frac{\phi_{1}(\Omega)}{2}\log d+\frac{\phi_{2}(\Omega)}{2}-\frac{3}{2}\right)+O_{\Omega,\varepsilon}\left(d^{5/6}\log^{7/6+\varepsilon}d\right).NEWLINE\]