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

摘要

Given a weight vector tau = (tau(1), ..., tau(n)) is an element of R-+(n) with each tau(i) bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set W-n(tau) boolean AND M, where M is a twice continuously differentiable manifold. From this we produce a lower bound for W-n(Psi) boolean AND M where psi 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.