7 problems

Let $\alpha$ be a real number whose Diophantine type is defined by the existence of a constant $c_\alpha>0$ such that … for all coprime integers $p$ and $q$. Consider the sequence…

Let $\alpha$ be a badly approximable real number, meaning that there exists a constant $c>0$ such that … for all rational numbers $p/q$. For a fixed integer $d\geq 2$, consider the…

For a real number $\alpha$, consider the sequence $(\{n^2\alpha\})_{n\ge1}$, where $\{x\}$ denotes the fractional part of $x$. Its pair correlation is Poissonian when it agrees wit…

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.

For a real number $\alpha$, consider the sequence $(\{n^2\alpha\})_{n\ge1}$, where $\{x\}$ denotes the fractional part of $x$. Its triple and higher correlations are called Poisson…

For a real number $\alpha>1$, write $\{\alpha^n\}$ for the fractional part of $\alpha^n$. The sequence $(\{\alpha^n\})_{n\ge1}$ has Poissonian $k$-point correlation when its $k$-po…