Rational approximations of real numbers

V. A. Ivanov

Matematičeskie zametki, Tome 23 (1978) no. 1, pp. 3-26

Résumé

For any $x\in\mathbf R$ put $$ c(x)=\varlimsup_{t\to\infty}\min_{\substack{(p,q)\in Z\times N\\q\le t}}t|qx-p|. $$ Let $[x_0;x_1,\dots,x_n,\dots]$ be an expansion of $x$ into a continued fraction and let $M=\{x\in J,\ \varlimsup\limits_{n\to\infty}x_n<\infty\}$. For $x\in M$ put $D(x)=c(x)/(1-c(x))$. The structure of the set $\mathfrak D=\{D(x),\ x\in M\}$ is studied. It is shown that $$ \mathfrak D\cap(3+\sqrt3,(5+3\sqrt3)/2)=\{D(x^{(n,3)})\}_{n=0}^\infty\nearrow(5+3\sqrt3)/2, $$ where $x^{(n,3)}=[\overline{3;(1,2)_n,1}]$. This yields for $\mu=\inf\{z,\mathfrak D\supset(z,+\infty)\}$ (“origin of the ray”) the following lower bound: $\mu\ge(5+3\sqrt3)/2=5,\!098\dots$. Suppose $a\in N$. Put $M(a)=\{x\in M,\ \varlimsup\limits_{n\to\infty}x_n=a\}$, $\mathfrak D(a)=\{D(x),\ x\in M(a)\}$. The smallest limit point of $\mathfrak D(a)$ $(a\ge2)$ is found. The structure of $\mathfrak D(a)$ is studied completely up to the smallest limit point and elucidated to the right of it.