Dimension estimates for badly approximable affine forms

For given ε>0 and b∈ Rᵐ , we say that a real m× n matrix A is ε -badly approximable for the target b if align*qⁿ, \|q\|→∞ \|q\|ⁿ Aq-bᵐ ≥ ε,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 ε -badly approximable matrices for fixed target b and the set of ε -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 ε -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.