Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 163 / 1 / c / Solution

Let $F(x)={\sum }_{n=1}^N{b}_{n}e(n^{2}x)$. Orthogonality givesThe number $r_2(m)$ of representations of $m$ as two squares is at most a constant times its divisor function. Since $m\le 2N^2$, the supplied divisor bound and Cauchy-Schwarz implyFor $2\le p\le 4$, monotonicity of $L^p$ norms on $[0,1]$ gives $\Vert F{\Vert }_p^p{\lesssim }_{ϵ}{N}^{ϵ}\Vert b{\Vert }_2^p$. For $p\ge 4$, use $\Vert F{\Vert }_{\infty }\le N^{1/2}\Vert b{\Vert }_2$ to obtainAfter enlarging the constant and the harmless epsilon loss, both ranges give

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