4.5 Primary estimate

Proposition 22 Primary estimate

✓

Set \(K_n:=2^{2d-n+1}(2\pi )^{-d}\). For all sufficiently large \(n\) and \(t\ge C_0 n^3\),

\[ K_n\operatorname {Re}\int _{\mathcal{D}_\mathrm {core}}\psi (\lambda )^{4t}\, d\lambda =\left(1-\frac{\binom {n}{3}}{8t} +O\! \left(\frac{n^2}{t}+\frac{n^{5/2}}{t^{3/2}}+\frac{n^6}{t^2} +e^{-cd}\right)\right)\hat A_{n,4t}. \]

Proof ▶

By the log-expansion (Lemma 19), on \(\mathcal{D}_\mathrm {core}\) we write \(\psi (\lambda )^{4t}=e^{-2t\| \lambda \| ^2}\, e^{-4itT}\, e^{4tQ}\, e^{4itP}\, e^{4tE_6}\) with \(|E_6|\le C_2 s^3\). Define intermediate integrals:

\[ M_{\mathrm{core}}:=\int _{\mathcal{D}_\mathrm {core}}e^{-2t\| \lambda \| ^2}e^{-4itT}e^{4tQ}e^{4itP}\, d\lambda , \qquad L_{\mathrm{core}}:=\int _{\mathcal{D}_\mathrm {core}}e^{-2t\| \lambda \| ^2}e^{-4itT}e^{4tQ}\, d\lambda . \]

Step 1: remove \(E_6\). \(|\psi ^{4t}-M_{\mathrm{core}}\cdot (\text{Gaussian})|\le |e^{4tE_6}-1|\le C t s^3\), and \(\int s^3 e^{-2t\| \lambda \| ^2}\le C(d/t)^3 F(d,t)\) by Gaussian radial moments.

Step 2: remove \(P\). \(|M_{\mathrm{core}}-L_{\mathrm{core}}|\le |e^{4itP}-1|\le 4t|P|\). By Cauchy–Schwarz and Lemma 18, \(\int |P|\, e^{-2t\| \lambda \| ^2}\le C n^{5/2}/t^{5/2}\cdot G_{\mathrm{core}}(d,t)^{1/2}\). This contributes \(O(n^{5/2}/t^{3/2})\hat A_{n,4t}\).

Step 3: remove \(Q\). \(|L_{\mathrm{core}}-\int e^{-2t\| \lambda \| ^2}e^{-4itT}|\le |e^{4tQ}-1|\) integrated against the Gaussian weight. Proposition 17 gives \(\int |e^{4tQ}-1|^2 e^{-2t\| \lambda \| ^2}\le C d^2/t^2\cdot G_{\mathrm{core}}(d,t)\), contributing \(O(d/t)=O(n^2/t)\) by Cauchy–Schwarz.

Step 4: cubic phase. By Lemma 21, \(\operatorname {Re}\int e^{-2t\| \lambda \| ^2}e^{-4itT}\ge [1-Cn^3/t]G_{\mathrm{core}}(d,t)\). The correction \(\binom {n}{3}/(8t)\) comes from the exact computation \(\mathbb {E}_{\gamma _t}[T^2]=\binom {n}{3}/(64t^3)\) (via the triangle formula) and \(1-\cos (4tT)\approx 8t^2T^2\).

Combining all four steps and using \(G_{\mathrm{core}}(d,t)=(1+O(e^{-cd}))F(d,t)\) (Lemma 11) yields the claimed expansion.