Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution

Zhiqin Tang; Qing-Long Zhou

number theory

Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution

arXiv:2607.10773

summary

The paper studies weak forms of inhomogeneous Poissonian pair correlation, showing they do not imply the full property or equidistribution, and demonstrates that different inhomogeneous parameters lead to independent notions of weak pair correlation.

Abstract

This paper investigates inhomogeneous Poissonian pair correlation (PPC), its weak form, and equidistribution. We establish that weak inhomogeneous PPC does not imply inhomogeneous PPC. Furthermore, we construct a sequence satisfying weak inhomogeneous PPC that fails to be equidistributed, which stands in sharp contrast to the homogeneous case. Finally, we prove that for distinct $I^_{1}^{3}, I^_{2}^{3} \in (0, \frac{1}{2}]$ , weak $I^_{1}^{3}$ -PPC does not imply weak $I^_{2}^{3}$ -PPC, showing that different inhomogeneous parameters give rise to mutually independent notions of weak inhomogeneous PPC.

Revised version with updated references and minor corrections

Topics & keywords

#poissonian pair correlation #inhomogeneous sequences #equidistribution #weak correlation #metric number theoryweak inhomogeneous PPCγ-parametersequence constructionpair correlationequidistribution