  In this paper, we study the $L^{p}$-improving property for the maximal
operators along a large class of curves of finite type in the plane with
dilation set $E \subset [1,2]$. The $L^{p}$-improving region depends on the
order of finite type and the fractal dimension of $E$. In particular, various
impacts of non-isotropic dilations are also considered.
