  We prove a shrinking target property for the action of subgroups of Veech
groups on square-tiled surfaces. Our main tool, of independent interest, is the
construction of a Fourier-like transform that we use to show that the
$L^2$-norm of the Koopman operator for the action is equivalent to the
$L^2$-norm of a Markov operator related to a random walk on a subgroup. This
generalizes work of V. Finkelshtein on the flat torus.
