Let E ⊂ [0, 1)^d be a set supporting a probability measure μ with
Fourier decay |u ̂(t)| ≪ (log |t|)^{−s} for some constant s > d + 1. Consider a
sequence of expanding integral matrices A=〖(A_n)〗_(n∈N) such that the minimal
singular values of A_(n+1) 〖A_n〗^(-1)are uniformly bounded below by K > 1. We
prove a quantitative Schmidt-type counting theorem under the following constraints:
(1) the points of interest are restricted to E; (2) the denominators of
the “shifted” rational approximations are drawn exclusively from A. Our result
extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to
the matrix setting, advancing the study of Diophantine approximation on fractals.
Moreover, it strengthens the equidistribution property of the sequence
〖(A_n X)〗_(n∈N) for μ-almost every x ∈ E. Applications include the normality of
vectors and shrinking target problems on fractal sets.
