On a lower bound of Hausdorff dimension of weighted singular vectors

Let.. = (..1,.,....) be a..-tuple of positive real numbers such that S...... = 1 and.. 1........ A..dimensional vector.. = (..1,.,....). R.. is said to be..- singular if for every.. > 0, there exists.. 0 > 1such that for all.. >.. 0, the system of inequalities max 1...... |...... -.... | 1.... <.... and 0 <.. <.. has an integer solution (..,..) = (..1,.,....,..). Z.. x Z. We prove that the Hausdorff dimension of the set of..singular vectors in R.. is bounded below by.. - 1 1+.. 1. Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in R2, J. Eur. Math. Soc. 22 (2020), 833-875].