The maximal-energy-concentration obstruction to Poissonian pair correlations

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 sequence $(\{a(n)\alpha\})_{n\geq1}$ on the torus. Pair-correlation obstruction conjecture. If $E(A_N)=\Omega(N^3)$, then there is no $\alpha$ for which the pair correlations of $(\{a(n)\alpha\})_{n\geq1}$ are Poissonian. This strengthens the known result that non-Poissonian pair correlations occur for every $\alpha$ in a positive-measure subset of $[0,1]$.