  Let $X=\text{SL}_3(\mathbb{R})/\text{SL}_3(\mathbb{Z})$, and
$g_t=\text{diag}(e^{2t}, e^{-t}, e^{-t})$. Let $\nu$ denote the push-forward of
the normalized Lebesgue measure on a segment of a straight line in the
expanding horosphere of $\{g_t\}_{t>0}$, under the map $h\mapsto
h\text{SL}_3(\mathbb{Z})$ from $\text{SL}_3(\mathbb{R})$ to $X$. We give
explicit necessary and sufficient Diophantine conditions on the line for
equidistribution of each of the following families of measures on $X$:
  (1) $g_t$-translates of $\nu$ as $t\to\infty$.
  (2) averages of $g_t$-translates of $\nu$ over $t\in[0,T]$ as $T\to\infty$.
  (3) $g_{t_i}$-translates of $\nu$ for some $t_i\to\infty$.
  We apply this dynamical result to show that Lebesgue-almost every point on
the planar line $y=ax+b$ is not Dirichlet-improvable if and only if
$(a,b)\notin\mathbb{Q}^2$.
