Dimension estimates for the set of points with non-dense orbit in homogeneous spaces (Q2193048)

Let \(\Gamma< G\) be a uniform lattice in a Lie group, and consider the action of a one-parameter subgroup \(\{g_t\mid{t\geqslant0}\}\) of diagonalizable elements acting by left-translation on the homogeneous quotient space \(X=G/\Gamma\). A standard argument using the uniqueness of the measure of maximal entropy for the action together with the variational principle shows that the Hausdorff dimension of the set of trajectories under the action that avoid a fixed open set is strictly smaller than the topological dimension of \(G\), and \textit{S. Kadyrov} [Dyn. Syst. 30, No. 2, 149--157 (2015; Zbl 1351.37013)] found effective bounds for the Hausdorff dimension of these sets of avoiding orbits. Here this result is extended to the setting of a lattice \(\Gamma\). Exponential mixing results for diagonalizable flows are used to find upper estimates for the Hausdorff dimension of the set of points whose orbits miss a subset of \(X\) with compact complement. This gives some new Diophantine applications including an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to \textit{R. Broderick} and the first author [Int. J. Number Theory 11, No. 7, 2037--2054 (2015; Zbl 1352.37008)].