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Source. Croot, published paper, p. 234, Theorem 1; proof on pp. 235–237.

Put $T(x)=\sqrt{\log x\log\log x}$.

Statement. For every $\eta>0$, for all sufficiently large $x$, any pairwise disjoint family $a_i\pmod{q_i}$ with distinct squarefree moduli $2\le q_1<\cdots<q_k\le x$ satisfies

$$k\le x\exp\left(-\left(\frac12-\eta\right)T(x)\right).$$

Complete relative proof. Let

$$B=\sqrt{\frac{\log x}{\log\log x}},\qquad Y=e^{T(x)}.$$

Discard the moduli with $\omega(q_i)\ge B$. By Lemma 2, their number is at most $x\exp(-(1/2+o(1))T(x))$. Also discard moduli all of whose prime divisors are at most $Y$. The external smooth-number estimate in Lemma 1 with $c=1$ bounds their number by the same expression.

Let $S_0$ be the remaining moduli. If it is empty, the two discarded-class bounds already suffice. Otherwise apply the complete selection lemma with parameters $B,Y$. It gives $Q=p_1\cdots p_t$ and a new prime $p>Y$. At least $|S_0|/(QB^{t+1})$ distinct integers at most $x$ are multiples of $Qp$. Therefore

$$\frac{|S_0|}{QB^{t+1}} \le\left\lfloor\frac{x}{Qp}\right\rfloor \le\frac{x}{Qp} <\frac{x}{QY}.$$

Cancel $Q$. Since $t+1<B$,

$$|S_0|<\frac{xB^{t+1}}Y\le\frac{xB^B}Y.$$

The logarithmic cost is

$$B\log B =\frac12\sqrt{\frac{\log x}{\log\log x}} \bigl(\log\log x-\log\log\log x\bigr) =\left(\frac12+o(1)\right)T(x).$$

Hence $|S_0|\le x\exp(-(1/2+o(1))T(x))$. Adding the two discarded-class bounds changes the exponential coefficient only by $o(1)$. For every fixed $\eta>0$, that error and the fixed multiplicative constants are eventually absorbed into $\eta T(x)$, proving the statement.

Source corrections and scope. The denominator $p_1\cdots p_k$ in one counting display on p. 236 should be $p_1\cdots p_t$. Non-strict finite counting inequalities are enough. The final estimate uses the new prime guaranteed by the corrected stopping rule in the linked selection lemma. The entire same-paper argument is included; only the exact analytic smooth-number theorem remains external.

Bears on. Problem 202: an upper bound with coefficient $1/2$ on the scale $T(x)$, only for families whose moduli are all squarefree; and the general-modulus upper bound, whose proof reduces to this theorem.