Abstract
In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions
involving averages over dilated submanifolds, thereby generalizing the theory
of spherical averages. Our main result is a quantitative estimate for the error
term of such averages valid for smooth functions under some effective mixing
assumptions on the action. With the aid of this theorem, we investigate
multiplicative-type Dirichlet-improvability for $(m\times n)$-matrices with
real coefficients. In particular, we establish that almost all matrices are
uniformly approximable by the function $x\mapsto x^{-1}(\log
x)^{-1+\varepsilon}$ for any $\varepsilon>0$. Results of this type motivate a
question which can be thought as a strengthening of Littlewood's conjecture in
multiplicative Diophantine approximation.