Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

The set of $Φ$ badly approximable matrices has full Hausdorff dimension (2312.14559v3)

Published 22 Dec 2023 in math.NT

Abstract: Recently Koivusalo, Levesley, Ward and Zhang introduced the set of simultaneously $\Phi$-badly approximable real vectors of $\mathbb{R}m$ with respect to an approximation function $\Phi$, and determined its Hausdorff dimension for the special class of power functions $\Phi(t)=t{-\tau}$. We refine this by naturally extending the formula to arbitrary decreasing functions, in terms of the lower order of $1/\Phi$ at infinity. We also provide an alternative, rather mild condition on $\Phi$ for this conclusion. Moreover, our results apply in the general matrix setting, and we establish an according formula for packing dimension as well. Thereby we also complement a recent refinement by Bandi and de Saxc\'e on the smaller set of exact approximation with respect to $\Phi$. Our basic tool is (a uniform variant of) the variational principle by Das, Fishman, Simmons, Urba\'nski. We also prove some new lower estimates regarding the set of exact approximation order in the matrix setting, which are sharp in special instances. For this we combine the result by Bandi and de Saxc\'e with a method developed by Moshchevitin.

Summary

We haven't generated a summary for this paper yet.