On the metric theory of multiplicative Diophantine approximation

In 1962, Gallagher proved a higher-dimensional version of Khintchine’s theorem on Diophantine approximation. Gallagher’s states that for any non-increasing approximation function ψ: ℕ → (0, 1/2) with $$\sum\nolimits_{q = 1}^\infty {\psi \left( q \right)} $$ and γ γ′ 0 the following set $$\left\{ {\left( {x,y} \right) \in {{\left[ {0,\,1} \right]}^2}:\left\| {qx - \gamma } \right\|\left\| {qy {\gamma ^\prime }} \right\| < \psi \right)\,\,{\rm{infinitely}}\,{\rm{often}}} \right\}$$ has full Lebesgue measure. Recently, Chow Technau fully inhomogeneous (without restrictions γ, γ′) above result. this paper, we prove an Erdős—Vaaler type result fibred multiplicative Along way, via different method, slightly weaker Chow—Technau’s condition at least one is not Liouville. We also extend Liouville fibres.