Factorized fourth moment for a bilinear quadratic exponential sum
= Factorized fourth moment for a bilinear quadratic exponential sum
Factoring $r_1^2-r_2^2=(r_1-r_2)(r_1+r_2)$ and similarly in $t$ separates diagonal contributions from a geometric sum. Grouping the three fixed factors by their product $m$ bounds the off-diagonal fourth moment by
$$
\sum_{m\ll RT^2}\tau_4(m)\min(R,\|\alpha m\|^{-1}).
$$