The invariance principle for inhomogeneous Diophantine approximations

Abstract

We establish the central limit theorem and the invariance principle for the inhomogeneous Diophantine approximations. The proof employs the cumulant method, which was developed by Björklund and Gorodnik to prove the central limit theorem in the homogeneous setting. Our approach also relies on the effective mixing of expanding translates for high-order correlations on the affine lattice space, extending the previous result by Kim.