Let d be a natural number and x an equidistributed sequence on the d-dimensional unit cube. In this talk, we give conditions implying the full measure of the set of points that can be approximated (with respect to a given approximation function) by the terms of x. For certain approximation functions, we determine the Hausdorff dimension whenever such a set is Lebesgue null. Our work relies on well-known discrepancy estimates as well as recent papers on ubiquitous systems by Wang and Wu (Mathematische Annalen (2021) 381:243–317) and Kleinbock and Wang (Advances in Mathematics 428 (2023) 109154).
This is joint work with M. Hussain, N. Shulga, and B. Ward.