= Bilinear cancellation for badly approximable phases
{title2=$|B|\ll DW\log^2X(D^{-1/2}+W^{-1/2})$}
Let $\alpha$ be a <badly approximable number>, and sum $B=\sum_{D<d\leq2D}\sum_{W<w\leq2W,dw\leq X}a_db_we(\alpha dw)$ with $|a_d|\leq\tau(d)$ and $|b_w|\leq1$. The <Cauchy-Schwarz inequality>, <divisor-square summatory bound> and geometric sums give $|B|^2\ll D\log^3X(DW+W\sum_{h\leq W}\min(D,\|h\alpha\|^{-1}))$. Separated rotations bound the last sum by $O_\alpha(W\log(2W))$, giving the displayed estimate. A hyperbolic cutoff remains valid because the allowed first-variable values after expansion form an interval.
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