4 problems

Tree-asymmetry conjecture. For every $v\in\{L,R\}^{\star}$,

Let $t\geq 1$ be an odd integer, and let $c_t$ denote the correlation quantity associated with the binary sum-of-digits difference for $t$. For each $K\geq 1$, consider the integer…

Cusick's conjecture. For all $t\geq 0$ we have

Cusick's simplified conjecture. For all $t\geq 0$, one has $c_t+c_{t'}>1$; equivalently, at least one of $t$ or $t'$ satisfies Cusick's conjecture. This is presented as an easier s…