Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 27 / 2 / a

The points are $\delta$-spaced if their circular spacing satisfies $\Vert {\theta }_r-{\theta }_s\Vert \ge \delta$ for $r\ne s$, where $\Vert t\Vert$ is distance to the nearest integer. Ordinary distance on the real line would be insufficient because the complex exponential is periodic.

Let $S(t)={\sum }_{M<n\le M+N}{a}_{n}e(nt)$ and $F(t)=e(-Mt)S(t)$. Multiplication by this unit-modulus factor leaves $|S(t)|$ unchanged and places the frequencies of $F$ in $1,\dots ,N$. Put $E=\sum |a_n{|}^2$. The permitted Sobolev–Gallagher inequality, in the form needed here, isFor $0<\delta \le 1$, the arcs about the ${\theta }_r$ have disjoint interiors on the circle group. Summing and applying the Cauchy-Schwarz inequality givesThe Cauchy-Schwarz inequality here follows by expanding $0\le \int |u-cv{|}^2$ and minimizing over $c\in \mathbb{C}$. For completeness, the finite-interval Parseval identities follow by expanding the squares: ${\int }_0^{1}e(kt)dt$ is one at $k=0$ and zero at every other integer $k$. Thus $\int |F{|}^2=E$ and $\int |F\prime {|}^2\le 4{\pi }^2{N}^{2}E$. We obtain the exponential-sum large sieve boundIf $\delta >1$, there is at most one point, and the direct Cauchy-Schwarz inequality bound $|S{|}^2\le NE$ proves the same assertion.