4 problems

Let $u$ be a binary word of length $n$ of the form $0^a1^{2^k}0^b$, where $a,b,k\in\mathbb{Z}_{\geq1}$. Power-of-two block conjecture. The word $u$ belongs to $\mathscr{U}$ if and…

Equidistribution conjecture. For every integer $N>1$ and every $a\in\mathbb{Z}/N\mathbb{Z}$,

Let $\pi:S[\{0,1\}]\to\mathbb{Z}_{\geq0}$ interpret a binary word as an integer, and let $u_1<u_2<\ldots<u_{k_n}$ be the ordered elements of $\pi(\mathscr{U}_n)$. Define … Let…

Ulam-word density conjecture. The density satisfies