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

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