  For a certain diagonal flow on $\operatorname{SL}(d, \mathbb{Z}) \backslash
\operatorname{SL}(d, \mathbb{R})$ where $d \geq 2$, we show that any bounded
subset (with measure zero boundary) of the horosphere or a translated
horosphere equidistributes, under a suitable normalization, on a target
shrinking into the cusp. This type of equidistribution is shrinking target
horospherical equidistribution (STHE), and we show STHE for several types of
shrinking targets. Our STHE results extend known results for $d=2$ and
$\mathcal{L} \backslash \operatorname{PSL}(2, \mathbb{R})$ where $\mathcal{L}$
is any cofinite Fuchsian group with at least one cusp.
  The two key tools needed to prove our STHE results for the horosphere are a
renormalization technique and Marklof's result on the equidistribution of the
Farey sequence on distinguished sections. For our STHE results for translated
horospheres, we introduce translated Farey sequences, develop some of their
geometric and dynamical properties, generalize Marklof's result by proving the
equidistribution of translated Farey sequences for the same distinguished
sections, and use this equidistribution of translated Farey sequences along
with the renormalization technique to prove our STHE results for translated
horospheres.
