Estimate (8.18) with sharpened constant: $L^\infty$ bound off the strongly major arc ($0.05$)

$L^\infty$ estimate off the strongly major arc, eq. (8.18), with constant $0.05$. Let $8.7\times10^{36}\le x\le e^{3100}$ be an integer, $K=1000$, $N_0=4\times10^{14}$, $T_0=3.29\times10^9$. For every $\alpha$ with $\|\alpha\|_{\mathbb R/\mathbb Z}>T_0/(3.6\pi x)$,

The paper proves this with $0.078$. The constant is governed by the crude regime $\alpha\ge20/N_0$, where $|S|\le1.01x/K$ and $|D|\le1/(2\alpha)$ give $1.01N_0/40\approx0.0253N_0=0.0757\,(N_0/3)$; the three Vinogradov-type regimes below $20/N_0$ are far smaller (the paper itself reports $<0.004$, $<0.013$, $<0.0002$). Moving the boundary to $\alpha\ge32/N_0$ makes the crude regime contribute $1.01N_0/64\approx0.0473\,(N_0/3)$, and for $\alpha<32/N_0$ the Dirichlet approximation $4\alpha=1/q+\mathcal O^*(1/q^2)$ has $q\ge N_0/128-1>3\times10^{12}\ge(x/K)^{1/3}$ whenever $x<3\times10^{40}$, so the intermediate regime (1.9) applies with bound $<0.013$; for larger $x$ the strongly minor regime (1.10) applies with $q>3\times10^{12}$ and bound $<0.0002$. The weakly minor regime via (1.12) gives $<0.018$ over the whole range $\log x\le3100$. Hence $0.05$ holds with room.

Same regimes and lemmas as the paper: symmetry (4.5); weakly major $T_0/(3.6\pi x)<\alpha\le KT_0/(4\pi x)$ via Proposition 7.2 and Lemma 3.3 with $\|\eta_0'\|_{L^1}=8\log2$ (5.11); crude regime via Lemma 4.3 with (5.11) and $|D_H(\alpha)|\le1/(2\|\alpha\|)$ from (2.1); (8.19)–(8.20) via Theorem 1.3. The sharper constant is needed because the companion $L^2$ bound is $7.4$, not $7.09$ (see minor_arc_L2_corrected): $0.05\times7.4=0.37<\tfrac23-0.1$.