Quasi-arithmetic sequences have no Poissonian pair correlation conjecture

Let $\left(a_n\right)_{n\geq 1}$ be a strictly increasing sequence of positive integers. It is quasi-arithmetic of degree $d$ if there exist constants $C,K>0$ and a strictly increasing sequence $\left(N_i\right)_{i\geq 1}$ of positive integers such that, for every $i\geq 1$, some subset $A^{(i)}\subset\left(a_n\right)_{1\leq n\leq N_i}$ with $\left|A^{(i)}\right|\geq CN_i$ is contained in a $d$-dimensional arithmetic progression of size at most $KN_i$, where $d$ is a positive integer. The quasi-arithmetic non-Poissonian pair correlation conjecture. If $\left(a_n\right)_{n\geq 1}$ is quasi-arithmetic of some degree $d$, then there is no $\alpha$ such that the pair correlation of $\left(\left\{a_n\alpha\right\}\right)_{n\geq 1}$ is Poissonian.

The paper states that this is equivalent to the maximal-additive-energy formulation because a sequence has additive energy $E(A_N)=\Omega(N^3)$ if and only if it is quasi-arithmetic of some degree. It strengthens the known full-measure result to exclude every value of $\alpha$.