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Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in$G$, let $U$ be an open subset of $X$, and let $\{g_t\}$ be a one-parametersubgroup of $G$. Consider the set of points in $X$ whose $g_t$-orbit misses$U$; it has measure zero if the flow is ergodic. It has been conjectured thatthis set has Hausdorff dimension strictly smaller than the dimension of $X$.This conjecture has been proved when $X$ is compact or when $G$ is a simple Liegroup of real rank $1$. In this paper we prove this conjecture for the case$G=\textrm{SL}_{m+n}(\mathbb{R})$, $\Gamma=\textrm{SL}_{m+n}(\mathbb{Z})$ and$g_t=\textrm{diag} (e^{nt}, \dots, e^{nt},e^{-mt}, \dots, e^{-mt})$, in factproviding an effective estimate for the codimension. The proof uses exponentialmixing of the flow together with the method of integral inequalities for heightfunctions on $\textrm{SL}_{m+n}(\mathbb{R})/\textrm{SL}_{m+n}(\mathbb{Z})$. Wealso discuss an application to the problem of improving Dirichlet's theorem insimultaneous Diophantine approximation.

On the dimension drop conjecture for diagonal flows on the space of lattices