3 problems

Let $r\geq 1$ be an integer. Let $b_1,\ldots,b_r$ be multiplicatively independent positive integers, and, for every $i$, $1\leq i\leq r$, let $\xi_i$ be a real number that is autom…

Let $p$ and $q$ be two multiplicatively independent natural numbers, and let automatic in base $b$ mean that the real number's expansion in base $b$ can be generated by a finite au…

Let $p$ and $q$ be two multiplicatively independent natural numbers, and let $x\in[0,1)$ be a real number. Define the forward orbit … where $T_q$ is the map on…