Effective pair correlations of fractional powers of complex grid points

Abstract

Using a standard definition of fractional powers on the universal cover $\exp:S\to \mathbb{C}^*$ seen as an infinite helicoid embedded in $\mathbb{R}^3$, we study the statistics of pairs from the countable family $\{n^\alpha \, : \, n \in \exp^{-1}(\Lambda) \}$ for every complex grid $\Lambda$ 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 $\Lambda$.