3 Basic Fourier estimates

Lemma 3.1

L^{2}

integral estimate

Let $ξ_{1}, \dots, ξ_{R}$ be real numbers that are $1 / N$ -separated. Then for any interval $I$ of length $T$ , and any sequence $a_{1}, \dots, a_{R}$ of complex numbers one has

\int_{I} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} d t = (T + O (N)) \sum_{r = 1}^{R} | a_{r} |^{2} .

Proof ▶

We adapt the proof of [ 113 , Theorem 9.1 ] . Without loss of generality we may normalize $\sum_{r = 1}^{R} | a_{r} |^{2} = 1$ . From the Plancherel identity we have

\int_{R} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} | \hat{ψ} ((t - t_{0}) / N) |^{2} d t = N

whenever $t_{0} \in R$ and $ψ$ is a smooth function supported on $[- 1 / 4, 1 / 4]$ of $L^{2}$ norm $1$ . By suitable choice of $ψ$ , this implies that

\int_{J} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} d t ≪ N

whenever $J$ is an interval of length $N$ . If one integrates 1 for all $t_{0} \in I$ , we see that

\int_{I} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} d t = T - \int_{R} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} (\int_{I} | \hat{ψ} ((t - t_{0}) / N) |^{2} d t_{0} - 1_{I} (t)) d t .

Since $\hat{ψ}$ is rapidly decreasing and has $L^{2}$ norm $1$ , one can compute

\int_{I} | \hat{ψ} ((t - t_{0}) / N) |^{2} d t_{0} - 1_{I} (t) ≪ (1 + dist (t, \partial I) / N)^{- 10}

and hence by 2 and the triangle inequality

\int_{R} | \sum_{r = 1}^{R} a_{r} e (ξ_{r} t) |^{2} (\int_{I} | \hat{ψ} ((t - t_{0}) / N) |^{2} d t_{0} - 1_{I} (t)) d t ≪ N

giving the claim. □