A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds

Abstract

Given a weight vector [Formula: see text] with each [Formula: see text] bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set [Formula: see text], where [Formula: see text] is a twice continuously differentiable manifold. From this we produce a lower bound for [Formula: see text] where [Formula: see text] is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in 2017, but we use an alternative mass transference style theorem proven by Wang, Wu and Xu (2015) to obtain our lower bound.