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

Table of contents
- Solution b

Solution

For each

τ \in Θ (r)

put

F_{τ} = \sum_{θ \in Θ (R) θ \subset τ} f_{θ} w_{r^{1/3}} .

Multiplication by

w_{r^{1/3}}

enlarges Fourier support by at most

r^{- 1/3}

because

w_{r^{1/3}}

is supported in

B_{r^{- 1/3}} (0)

. The supports of the

F_{τ}

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

\int_{R^{3}} \sum_{θ \in Θ (R)} f_{θ} w_{r^{1/3}}^{2} ≲ \sum_{τ \in Θ (r)} \int_{R^{3}} \sum_{θ \in Θ (R) θ \subset τ} f_{θ} w_{r^{1/3}}^{2} .

Solved by gpt-5.6-sol high.

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