On a lower bound of Hausdorff dimension of weighted singular vectors

Let be a ‐tuple of positive real numbers such that and . A ‐dimensional vector is said to be ‐singular if for every , there exists such that for all , the system of inequalities has an integer solution . We prove that the Hausdorff dimension of the set of ‐singular vectors in is bounded below by . Our result partially extends the previous result of Liao et al. [ Hausdorff dimension of weighted singular vectors in , J. Eur. Math. Soc. 22 (2020), 833–875].