9 problems

Quadratic-scaling pair correlation conjecture. A nonzero asymptotic pair correlation density exists for the quadratic scaling. Moreover, it exhibits strong level repulsion and vani…

Larcher–Stockinger conjecture. If for almost all $\alpha$ the pair correlations of $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ are not Poissonian, then the pair correlations of this se…

Let $\mathcal{A}\subset\mathbb{N}$ be an increasing integer sequence. Say that $\mathcal{A}$ has MPPC if its homogeneous pair correlations are Poissonian, and let … holds for…

Almost-everywhere obstruction conjecture. If for almost all $\alpha$ the pair correlations of $((\lbrace a_n\alpha\rbrace)_{n\in\mathbb N})$ are not Poissonian, then the pair corre…

Let $(a(n))_{n\geq1}$ be a strictly increasing sequence of positive integers. Call it quasi-arithmetic if it has the finite-dimensional arithmetic-progression structure defined in…

Let $(a(n))_{n\geq1}$ be the sequence of integers used to form $A_N=\{a(1),\ldots,a(N)\}$, and let $E(A_N)$ denote its additive energy. For a real number $\alpha$, consider the seq…

Let $d\geq 2$, let $\alpha$ be a real number, and write $\langle t\rangle$ for the fractional part of $t$. The sequence $\{\langle\alpha x^d\rangle\}_{x\geq 1}$ has Poissonian pair…

Let $\alpha$ be irrational and let $d\geq2$, and consider the sequence $\alpha n^d$ modulo one. A real number $\alpha$ is badly approximable by rationals if its rational approximat…

Linear pair-correlation conjecture. For almost every $\lambda\in(\frac{1}{2},1)$, there exist $c,C>0$ such that