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 {\...

In conventional gravitational wave scattering theory, a large-distance asymptotic approximation is employed. this approximation, the approximated by its asymptotics. paper, we establish theory without approximation.