Which structure does the set of numbers (possibly rational) with the property, that its continued fraction only contains the numbers $1$ and $2$, have ?
Let us denote the set of such real numbers with $S$.
$S$ contains numbers as $1$ , $2$ , $\ \sqrt{2}+1\ $ and $\ \frac{\sqrt{5}+1}{2}\ $. Since the set of infinite sequences with entries $1$ or $2$ is uncountable, there are uncountable many irrational numbers in $S$. Which elements of $S$ are limit points ? Is $S$ dense in a real interval ?
To avoid ambiguity, a number like $[1,2,1,1,2,2,1]$ is NOT in $S$ since it can be written as $[1,2,1,1,2,3]$, which has a "$3$".