The exponential sum S(α; N) = ∑_{n≤N} b(n) e(n² α) satisfies S(α;N) / (N/√log N) ≪ N^ε (q^{-1/4} + N^{-1/2} q^{1/4} + N^{-1/8}) for α near a/q with (a,q)=1.
A new form of the circle method, and its application to quadratic forms
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Bounding the exponential sum on squares of some sifted sequences
The exponential sum S(α; N) = ∑_{n≤N} b(n) e(n² α) satisfies S(α;N) / (N/√log N) ≪ N^ε (q^{-1/4} + N^{-1/2} q^{1/4} + N^{-1/8}) for α near a/q with (a,q)=1.