Abstract

Using a standard definition of fractional powers on the universal cover $\exp <!--FUNCTION\; APPLICATION-->:S\to {ℂ}^{\ast }$, where $S$ is the standard infinite helicoid embedded in ${ℝ}^3$, we study the statistics of pairs at various scalings from the countable family $\{n^{\alpha }:n\in {\exp <!--FUNCTION\; APPLICATION-->}^{-1}(\Lambda )\}$ for every complex grid $\Lambda$ and every real parameter $\alpha \in ]0,1[$. We prove the convergence of the empirical pair correlation measures towards a rotation-invariant measure with explicit density. In particular, with the scaling factor $N\mapsto N^{1-\alpha }$, 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. We give an error term for this convergence.

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