Tao Theorem 5.1: the Type I estimate (with corrected constants)

Let $q\ge2$, let $a$ be coprime to $q$, and let $4\alpha=\frac aq+\beta$ with $|\beta|\le q^{-2}$. Let $x>0$, let $L,C\ge0$, and let $M\ge q/2$. Suppose a nonnegative weight $W$ obeys, for every $1\le d\le M$, the pointwise bound

(with the value $\tfrac12\frac xdL+C$ where the sine vanishes). Then

This is the Type I estimate of the source's minor-arc theorem. In the application $M=UV$ is the length of the divisor range, $L=\log x$, $C=4\log2\log 2x$, and $W(d)$ is the modulus of the inner exponential sum $\bigl|\sum_n(\log n+c_d\log d)\eta_0(dn/x)e(\alpha dn)\bigr|$, whose pointwise bound is what Corollary 3.2 supplies. The two pieces are the small divisors $d\le q/2$, where the frequency $4d\alpha$ is bounded away from the integers so the reciprocal-sine weight never exceeds $2q$, and the remaining range, cut into blocks of length $2q$ on each of which the weight $x/d$ is frozen at the left endpoint and the odd-restricted Vinogradov-type lemma is applied once.

Deviation from the source The source's corresponding display is

which its own chain does not give, for two independent reasons. First, the integral test it invokes drops an additive constant: the $j=0$ term of $\sum_j\frac{x}{2jq+q/2}$ alone equals $\frac{x}{2q}\cdot4$, so the claimed bound $\frac{x}{2q}\log(\frac{2UV}q+4)$ fails whenever $UV/q<\frac{e^4-4}2\approx25.3$ (at $q=1$, $UV=10$ the two sides are $2.8937x$ and $1.5890x$). Second, its per-block application of Corollary 3.5 uses the factor $1$ where the corollary gives $\lfloor\frac{2q}{2q}\rfloor+1=2$, the blocks having length exactly $2q$. The bound above is what the chain actually yields: the logarithm gains the additive $4$, the leading coefficient is $1$ rather than $\tfrac12$, and the block constant doubles. Nothing here contradicts the source's Theorem 5.1, whose statement carries further terms; only the intermediate Type I display is affected.

Formalization Note The divisor range is the odd integers of $(0,\lfloor M\rfloor]$. The truncation parameters are kept abstract as $L$ and $C$ rather than specialized to $\log x$ and $4\log2\log2x$. At the zeros of the sine the weight takes the value of the first argument of the minimum, which is the source's convention. The Vinogradov-type lemma of the source (its Lemma 3.4) is carried as the hypothesis hvino, quantified over the truncation level because the blocks use different ones; the odd-restricted Corollary 3.5 is imported and applied to it.