= Weyl inequality
{c}
{title2=$|S|\ll_{k,\eta}N^{1+\eta}(q^{-1}+N^{-1}+qN^{-k})^{2^{1-k}}$}
For a real degree-$k$ polynomial with leading coefficient $\theta$, $k\ge2$, and coprime integers $a,q$ satisfying $|\theta-a/q|\le q^{-2}$, the displayed bound applies to its <exponential sum> over $1\le n\le N$. Repeated <Weyl differencing> reduces the phase to slope $k!\theta h_1\cdots h_{k-1}$. A geometric-sum estimate, the <subpower bound for the divisor function> and the <reciprocal fractional-part sum near a rational> complete the estimate. The arbitrary positive exponent $\eta$ absorbs divisor bounds and logarithms.
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