In this talk, we give conditions entailing full measure of sets of points in n dimensional Euclidean space that can be approximated (with respect to a given approximation function) by the terms of a uniformly distributed sequence. We also determined their Hausdorff dimension whenever such sets are null for certain approximation functions. Our work relies on previously known discrepancy estimates as well as recent papers by Wang and Wu (Mathematische Annalen (2021) 381:243–317) and Kleinbock and Wang (Advances in Mathematics 428 (2023) 109154) on ubiquitous systems defined on rectangles.
This is joint work with M. Hussain, N. Shulga, and B. Ward. See arXiv:2308.16603 [math.NT]
