4 problems

Digit-sum strengthening. For every integer $k\geq 9$,

Maximal subword-occurrence conjecture. The same equality should hold for any finite word $w$:

For an integer base $b\ge 2$, let $A_b(n)$ denote the sequence counting the cumulative number of distinct subwords occurring in the base-$b$ expansions of the integers up to $n$, a…

Digit-discrepancy conjecture. The number of unequal digits between $Z_b(b^k)$ and the first $k$ digits of the base-$b$ expansion of $\vartheta(b)$ is smaller than or equal to the n…