Continued fractions with odd partial quotients (Q2909603)

Every rational number \(x\in[0,1]\) can be uniquely represented as ``odd'' continued fraction NEWLINE\[NEWLINEx=1+\dfrac{\varepsilon_1}{a_1+\dfrac{\varepsilon_2}{a_2+ {\atop\ddots\,\displaystyle{+\dfrac{\varepsilon_l}{a_l}}}}},NEWLINE\]NEWLINE where all \(a_i\) are odd, \(\varepsilon_i=\pm1\) (\(\varepsilon_1=-1\)) and \(a_j+\varepsilon_{j+1}\geq 2\) for \(j\geq 1\). (If \(a_l=1\), then \(\varepsilon_l=1\) for uniqueness of the representation.)NEWLINENEWLINELet NEWLINE\[NEWLINES(x)=\sum\limits_{j=1}^{l}a_jNEWLINE\]NEWLINE be a sum of all partial quotients of \(x\) and NEWLINE\[NEWLINEM_n=\left\{x\in\mathbb{Q}\cap[0,1]:S(x)\leq n+1\right\}.NEWLINE\]NEWLINE The author studies the limit distribution function NEWLINE\[NEWLINEF(x)=\lim\limits_{n\to\infty}F_n(x),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEF_n(x)=\frac{\#\left\{\xi\in M_n:\xi\leq x\right\}}{\# M_n},\qquad(x\in[0,1]).NEWLINE\]NEWLINE The function \(F\) plays the same role for odd continued fractions as Minkowski question mark function plays for classical continued fractions. The paper under review describes main properties of the function \(F\). In particular it is proved that this function is singular and satisfies a number of functional equations.