3 problems

Prime-rotation conjecture. The prime rotation sequence $(p_n\alpha)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders and, consequently, has Poissonian gap distr…

Monotonicity conjecture. If $x$ has $(k,\beta_2)$-Poissonian box correlations, then it also has $(k,\beta_1)$-Poissonian box correlations.

Let $m\ge 2$, let $\alpha>0$, and let $0<\theta<1/m$. Consider the sequence of fractional parts $(\alpha n^\theta\bmod 1)_{n>0}$. Poissonian correlation conjecture. For every such…