Effective statistics of pairs of fractional powers of complex grid points

Using a standard definition of fractional powers on the universal cover exp:S→C∗, where S is the standard infinite helicoid embedded in R3, we study the statistics of pairs at various scalings from the countable family {nα:n∈exp−1(Λ)} for every complex grid Λ and every real parameter α∈]0,1[. We prove the convergence of the empirical pair correlation measures towards a rotation-invariant measure with explicit density. In particular, with the scaling factor N↦N1−α, 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. We give an error term for this convergence.