Alladi Ramakrishnan Hall

Diophantine approximation with nonsingular integral transformations

Sreekrishna Dani

CEBS, Mumbai

Let $V$ be the Cartesian product of $p$ copies of the Euclidean space $\R^n$, where $p \leq n-1$ and consider the component-wise action on $V$ by the multiplicative semigroup, say $\Gamma$ of nonsingular $n \times n$
matrices with integer entries. Then for $v=(v_1, \dots, v_p)\in V$ the orbit is dense in $V$ if and only if no nontrivial linear combination of $v_1, \dots, v_p$ is a rational vector; thus given a $v$ satisfying the
condition and $w\in V$, for all $\epsilon >0$ there exists $\gamma \in \Gamma$ such that $|| \gamma v -w ||<\epsilon$. We discuss the effectiveness of the approximation, namely the issue of what bound we can
have on the size of $\gamma$ in terms of $\epsilon$. The exponent of approximation associated with the orbit turns out to be $(n-p)/p$ for all initial points $v$ outside a null set, which is described by a certain
Diophantine condition.