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Source. Croot, published paper, p. 235, Lemma 2. The factorial estimate compressed in the source is expanded here, including the endpoint needed in Theorem 1.

Write

$$\omega(n)=\#\{p:p\text{ prime},\ p\mid n\},\quad B(x)=\sqrt{\frac{\log x}{\log\log x}},\quad T(x)=\sqrt{\log x\log\log x}.$$

Statement. For each fixed $c>0$,

$$\#\{n\le x:\omega(n)\ge cB(x)\} \le x\exp\left(-\left(\frac c2+o(1)\right)T(x)\right).$$

In particular this bounds the strict inequality in the printed lemma too.

Complete proof. Set $r=\lceil cB(x)\rceil$. For $x$ sufficiently large, $r\ge1$. An integer with $\omega(n)\ge r$ contains a set of $r$ distinct prime divisors. Counting such sets gives

$$\begin{aligned} \#\{n\le x:\omega(n)\ge r\} &\le\sum_{n\le x}\binom{\omega(n)}r\\ &=\sum_{p_1<\cdots<p_r\le x} \left\lfloor\frac{x}{p_1\cdots p_r}\right\rfloor\\ &\le\frac{x}{r!}\left(\sum_{p\le x}\frac1p\right)^r. \end{aligned}$$

The elementary reciprocal-prime estimate is $A(x)=O(\log\log x)$, and

$$\log(r!)=\sum_{j=1}^r\log j \ge\int_1^r\log t\,dt=r\log r-r+1.$$

Consequently the logarithm of the factor multiplying $x$ is at most

$$-r\log r+r\log A(x)+r =-\left(\frac c2+o(1)\right)T(x).$$

Indeed, $r=(c+o(1))B(x)$, $\log r=\tfrac12\log\log x+O(\log\log\log x)$, and $\log A(x)=O(\log\log\log x)$. Thus $r\log r=(c/2+o(1))T(x)$, while both remaining terms are $o(T(x))$.

Source clarification. No estimate that is uniform in a growing $c$ is used. The weak threshold above also covers the discarded $\omega(n)\ge B(x)$ class in Theorem 1, even when $B(x)$ is an integer.

Bears on. Theorem 1 and Problem 202.