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 editorsSayous, Rafael

Journal or seriesCombinatorics and Number Theory

ISSN2996-2196

eISSN2996-220X

Publication year2025

Publication date10/12/2024

Volume14

Pages range13-47

PublisherMathematical Sciences Publishers

Publication countryUnited States

Publication languageEnglish

DOIhttps://doi.org/10.2140/cnt.2025.14.13

Publication open accessNot open

Publication channel open access

Publication is parallel publishedhttps://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.


Keywordsnumber theorylattice theorymeasure theory

Free keywordspair correlations; level repulsion; fractional power; lattices; convergence of measures


Contributing organizations


Ministry reportingYes

VIRTA submission year2025

Preliminary JUFO rating1


Last updated on 2025-25-01 at 20:06