Submanifold-genericity of ${\mathbb{R}}^d$-actions and uniform multiplicative Diophantine approximation

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 $x\mapsto x^{-1}(\log x{)}^{-1+\epsilon }$ for any $\epsilon >0$. Results of this type motivate a question which can be thought as a strengthening of Littlewood’s conjecture in multiplicative Diophantine approximation.