Dimension estimates for badly approximable affine forms

Abstract For given >0 and b Rᵐ, we say that a real m n matrix A is -badly approximable for the target b if align*ₐ䂞, \|ₐ\| \|q\|ⁿ Aq-bᵐ, align* where denotes the distance from the nearest integral vector. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of -badly approximable matrices for fixed target b and the set of -badly approximable targets for fixed matrix A. Moreover, we give a Diophantine condition of A equivalent to the full Hausdorff dimension of the set of -badly approximable targets for fixed A. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the A -fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.