Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 163 / 4 / b / Solution

For each $\tau \in \Theta (r)$ putMultiplication by $w_{r^{1/3}}$ enlarges Fourier support by at most $r^{-1/3}$ because ${\hat{w}}_{r^{1/3}}$ is supported in $B_{r^{-1/3}}(0)$. The supports of the $F_{\tau }$ therefore have uniformly bounded overlap: separated $r^{-1/3}$-scale moment-curve intervals remain disjoint except for boundedly many neighbors.

The fourier-support almost orthogonality supplied by the Plancherel theorem now gives