Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution

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 ${\gamma }_1,{\gamma }_2\in (0,\frac{1}{2}]$, weak ${\gamma }_1$-PPC does not imply weak ${\gamma }_2$-PPC, showing that different inhomogeneous parameters give rise to mutually independent notions of weak inhomogeneous PPC.