Effective pair correlations of fractional powers of complex grid points

Using a standard definition of fractional powers on the universal cover $\exp :S\to {\mathbb{C}}^{\ast }$ seen as an infinite helicoid embedded in ${\mathbb{R}}^3$, we study the statistics of pairs from the countable family $\{n^{\alpha }:n\in {\exp }^{-1}(\mathrm{\Lambda })\}$ for every complex grid $\mathrm{\Lambda }$ and every real parameter $\alpha \in ]0,1[$. We prove the convergence of the empirical pair correlations measures towards a rotation invariant measure with explicit density. In particular, with the scaling factor $N\mapsto N^{1-\alpha }$, we prove that there exists an exotic pair correlation function which exhibits a level repulsion phenomenon. For other scaling factors, we prove that either the pair correlations are Poissonian or there is a total loss of mass. In addition, we give an error term for this convergence, with explicit dependence on parameters of the grid $\mathrm{\Lambda }$.