. For given ǫ ą 0 and b P R m , we say that a real m ˆ n matrix A is ǫ -badly approximable for the target b if lim inf q P Z n , } q }Ñ8 } q } n x Aq ´ b y m ě ǫ, where x¨y denotes the distance from the nearest integral vector. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of ǫ -badly approximable matrices for fixed target b and the set of ǫ -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 ǫ -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.
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