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Extremal manifolds and Hausdorff dimension

The recent proof by D. Y. Kleinbock and G. A. Margulis [11] 
of Sprindžuk’s conjecture for smooth nondegenerate manifolds M means that the set Lv(M) of v-approximable points (this and other terminology is explained below) on M is of zero induced Lebesgue measure. This raises the question of its Hausdorff dimension.