Dr Mumtaz Hussain


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Weighted approximation for limsup sets

We just uploaded to the arXiv a paper concerning “weighted approximation for limsup sets” jointly with Gerardo Gonzalez Robert, Nikita Shulga, and Ben Ward. The link to the arXiv is https://arxiv.org/abs/2308.16603.

Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine ap-
proximation are fundamental results on the metric properties of Ψ-well approximable sets.
These foundational results have since been generalised to the framework of weighted Dio-
phantine approximation for systems of real linear forms (matrices). In this article, we prove
analogues of these weighted results in a range of settings including the p-adics (Theorems
7 and 8), complex numbers (Theorems 9 and 10), quaternions (Theorems 11 and 12), and
formal power series (Theorems 13 and 14). We also consider approximation by uniformly
distributed sequences. Under some assumptions on the approximation functions, we prove
a 0-1 dichotomy law (Theorem 15). We obtain divergence results for any approximation
function under some natural restrictions on the discrepancy (Theorems 16, 17, and 19).
The key tools in proving the main parts of these results are the weighted ubiquitous
systems and weighted mass transference principle introduced recently by Kleinbock and
Wang [Adv. Math. 428 (2023), Paper No. 109154], and Wang & Wu [Math. Ann. 381 (2021),
no. 1-2, 243–317] respectively.

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