Minor arc bound (8.17) for the five-primes circle method

The minor arc bound in Tao's five-primes paper, equation (8.17).

The constants and notation are those of Section 8 of the paper: $N_0 = 4\times10^{14}$, $K = 10^3$, $T_0 = 3.29\times10^9$, the cutoffs $\eta_1(t) = (1-10\,\mathrm{dist}(t,[0.2,0.8]))_+$ and $\eta_0(t) = 4(\log 2 - |\log 2t|)_+$, the smoothed prime exponential sums $S_{\eta,q}(y,\alpha) = \sum_n \Lambda(n)\mathbf 1_{(n,q)=1}\eta(n/y)e(n\alpha)$, and the Dirichlet kernel $D_H(\alpha) = \sum_{1\le n\le H}e(n\alpha)$. Let $x$ be an integer with

Then, with $M = \frac{x^2}{K}\lfloor N_0/3\rfloor^{3}$,

This is the complement of Proposition 8.3. That proposition puts the strongly major arc contribution within $0.1M$ of $\frac23 M$; since $\frac23 - 0.1 - 0.56 = \frac{1}{150} > 0$, the two estimates together force the weighted representation count $R(x,N_0)$ to be strictly positive, which is exactly Theorem 8.2. In the paper this bound is in turn obtained from Proposition 8.4, the $L^2$ estimate $\int_{\text{minor}}|S_{\eta_1}|^2|D_{N_0/3}|^2 \le 7.09\,(N_0/3)^2x$, together with the $L^\infty$ estimate (8.18), by Hölder's inequality.

A remark on the source. As printed, the right-hand side of (8.17) reads $0.56\frac{x^2}{K}(N_0/3)^2$. The exponent $2$ there is a misprint for $3$, and the statement above uses $3$. Two independent checks fix it: the quantity being compared against is Proposition 8.3's main term $\frac{x^2}{K}(N_0/3)^3(\frac23 + \mathcal O^*(0.1))$, so the comparison is only meaningful at $(N_0/3)^3$; and the paper's own derivation multiplies Proposition 8.4's $7.09\,(N_0/3)^2x$ by the $L^\infty$ bound of size $0.078\,(N_0/3)\frac{x}{K}$, giving $0.553\,\frac{x^2}{K}(N_0/3)^3$, which is what $0.56$ rounds. With the printed exponent $2$ the inequality would be a far stronger claim that the paper neither proves nor needs. (The display (8.18) carries a matching slip, an extra factor of $x$ on its right-hand side.)

Formalization Note The circle is AddCircle 1 with its Haar probability measure and $\|\alpha\|_{\mathbb R/\mathbb Z}$ is its norm, the distance to the nearest integer. The integrand is the norm of the published representationIntegrand: since $|e(-x\alpha)| = 1$, that norm is exactly $|S_{\eta_1}|^2|S_{\eta_0}||D_{N_0/3}|^3$, the three finite Fourier polynomials agreeing with the corresponding sums because $\eta_1$ is supported in $[0.1,0.9]$ and $\eta_0$ in $(1/4,1)$.

The domain is the complement of the closed ball of radius $T_0/(3.6\pi x)$, that is the open condition $\|\alpha\| > T_0/(3.6\pi x)$, rather than the paper's closed $\ge$; the two domains differ by two points, a null set, and in any case the open one is contained in the closed one, so this statement is implied by the printed inequality. As in Proposition 8.3, $\lfloor N_0/3 \rfloor$ is used for the Dirichlet kernel range and in $M$.