5.3 Residual estimate

Proposition 25 Residual estimate

✓

Set \(\delta ^2=2d/t\) and \(K_n:=2^{2d-n+1}(2\pi )^{-d}\). Then

\[ \begin{aligned} K_n\int _{\mathcal{B}_\delta \setminus \mathcal{D}_\mathrm {core}}|\psi (\lambda )|^{4t}\, d\lambda & = o(\hat A_{n,4t}),\\ (2\pi )^{-d}\left|\int _{R_\delta }\psi (\gamma )^{4t}\, d\gamma \right| & = o(\hat A_{n,4t}), \end{aligned} \]

as \(n\to \infty \) with \(t/(n^{8/3}\log t)\to \infty \). For \(t\ge C_0 n^3\) and all sufficiently large \(n\), both bounds are at most \(Ce^{-cn^2}\hat A_{n,4t}\).

Proof ▶

Outside the core: If \(\lambda \in (\mathcal{B}_\delta \cap \mathcal{D}_r)\setminus \mathcal{D}_\mathrm {core}\), then \(\| \lambda \| ^2{\gt}d/t\), so Lemma 20 gives \(|\psi |^{4t}\le e^{-d/2}e^{-t\| \lambda \| ^2}\), and the integral is \(O(e^{-cd})G_{\mathrm{core}}(d,t)\).

Far shell: Corollary 24 bounds the contribution of \(\mathcal{B}_\delta \setminus \mathcal{D}_r\).

Residual \(R_\delta \): Decompose into even cells (handled by small-ball decay + far-shell bounds) and odd cells (Lemma 5 gives \(|\psi |^{4t}\le 2^{-2t}\)).

At the cubic threshold \(t\ge C_0 n^3\), all bounds are \(O(e^{-cn^2})\hat A_{n,4t}\).