9 problems

Uniform frequency conjecture. For almost all $x \in \mathbb{C}$, any two words of the same length in $D^*$ appear with the same frequency in the $\alpha$-expansion of $x$.

Let $b\geq 2$ be an integer, and let a number have an expansion in base $b$ whose digit sequence is automatic. Cobham's conjecture. Every such number is either rational or transcen…

Let $b\ge 2$ be an integer, and let $x$ be an irrational real number. Its base-$b$ expansion is the digit sequence obtained by writing $x$ in base $b$. A sequence is automatic if i…

For each integer base $b\geq 2$, let $T_b$ be the set of numbers in $(0,1)$ whose continued fraction expansion $[0;a_1,a_2,\dots]$ equals their base-$b$ expansion…

Algebraic-irrational complexity conjecture. For every algebraic irrational $x$, the complexity function satisfies

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 $n$ be an integer. Erdős's conjecture. The digit $2$ occurs in the ternary expansion of $2^n$ for all $n>8$. This is a specific conjecture about the digit structure of powers o…

For $alphain[1,2]$, let $mualpha$ be the invariant measure associated with the signed binary expansion map $Salpha$ on $[-1,1]$. The digit $0$ occurs precisely when an iterate lie…