The invariance principle for inhomogeneous Diophantine approximations

Songzi Li

The invariance principle for inhomogeneous Diophantine approximations

Probability 2026-06-30 v1

Authors: Songzi Li

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 Bj\"orklund 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.

Keywords

inhomogeneous Diophantine approximation central limit theorem invariance principle cumulant method effective mixing affine lattice

Cite

@article{arxiv.2606.31008,
  title  = {The invariance principle for inhomogeneous Diophantine approximations},
  author = {Songzi Li},
  journal= {arXiv preprint arXiv:2606.31008},
  year   = {2026}
}