A1 Journal article (refereed)
Effective statistics of pairs of fractional powers of complex grid points (2025)
Sayous, R. (2025). Effective statistics of pairs of fractional powers of complex grid points. Combinatorics and Number Theory, 14, 13-47. https://doi.org/10.2140/cnt.2025.14.13
JYU authors or editors
Publication details
All authors or editors: Sayous, Rafael
Journal or series: Combinatorics and Number Theory
ISSN: 2996-2196
eISSN: 2996-220X
Publication year: 2025
Publication date: 10/12/2024
Volume: 14
Pages range: 13-47
Publisher: Mathematical Sciences Publishers
Publication country: United States
Publication language: English
DOI: https://doi.org/10.2140/cnt.2025.14.13
Publication open access: Not open
Publication channel open access:
Publication is parallel published: https://doi.org/10.48550/arXiv.2310.19578
Abstract
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.
Keywords: number theory; lattice theory; measure theory
Free keywords: pair correlations; level repulsion; fractional power; lattices; convergence of measures
Contributing organizations
Ministry reporting: Yes
VIRTA submission year: 2025
Preliminary JUFO rating: 1