  In this paper, we give a new proof for the Hausdorff dimension of the
non-dense orbit set for expanding maps. This proof is based on the sharp lower
bound of the Hausdorff dimension of repellers given by Cao, Pesin and Zhao
in~\cite{caopesinzhao2019}. Besides, we prove that the
Carath$\acute{\text{e}}$odory singular dimension of non-dense orbit set equals
that of the repeller.
