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Source. Lemma 2, attributed to Croot, printed p. 144 (PDF p. 2). For every fixed $c>0$, the number of positive integers $n\le x$ with

$$\omega(n)>c\sqrt{\frac{\log x}{\log\log x}}$$

is at most

$$x\exp\left(-\left(\frac c2-o(1)\right) \sqrt{\log x\log\log x}\right).$$

Here $\omega(n)$ counts distinct prime divisors, with $\omega(1)=0$.

The complete proof is Croot’s canonical Lemma 2 reconstruction. Its weak-threshold estimate also bounds Chen’s strict-threshold set. This is an imported, already reconstructed proof; it is not duplicated or counted as another proof component here. Chen uses it in Lemma 3 to separate distinct-prime growth from repeated prime factors.