The set of symmetrically $Φ$ badly approximable matrices has full Hausdorff dimension

Johannes Schleischitz

Published 2023-12-22Version 1

Given a decreasing approximation function $\Phi$, in a recent paper by Koivusalo, Levesley, Ward and Zhang simultaneously $\Phi$-badly approximable real vectors were introduced. An upper bound for its Hausdorff dimension can be easily obtained in terms of the lower order of $\Phi$ at infinity. In the aforementioned paper, for power functions $\Phi$ the reverse lower bound was obtained to settle the exact value of the dimension. An improvement in terms of showing the same result for the smaller sets of real vectors with ``exact'' approximation order $\Phi$, for arbitrary decreasing $\Phi$, was obtained very recently by Bandi and de Saxce. We introduce a twisted, slightly larger set of symmetric $\Phi$ badly approximable real vectors (matrices). We use the variational principle by Das, Fishman, Simmons, Urbanski to show that the analogous full dimension result remains true for a larger class of $\Phi$ that are not necessarily monotonic upon another rather mild decay condition, meaning then our symmetric $\Phi$ approximable vectors have full dimension. Moreover our result holds in the general matrix setting and we establish the corresponding result for packing dimension as well.