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Statement

For sufficiently large $N$, $\xi\in(0,1)$, and $A\subseteq[1,N]$ with $R(A)\ge\eta$, there is $A'\subseteq A$ such that

$$R(A')\ge(1-\xi)\eta,\qquad qR(A'_q)\ge\frac{\eta\xi}{2\log\log N}\quad(q\in\mathcal Q_{A'}).$$

Here $A_d=\{n\in A:d\mid n\}$ and $\mathcal Q_A$ is the set of prime powers dividing some element of $A$.

Source. Liu–Sawhney, arXiv:2404.07113v1, Lemma 6.2, p. 19. The denominator is printed $2\log\log n$ in the statement and first proof line, with an unbound lowercase $n$; the final displayed estimate uses $N$, as written here.

Rewritten proof

From the current set, delete all multiples of any prime power $q$ whose fiber has mass less than $\eta\xi/(2q\log\log N)$. Every deletion removes at least one integer, so the process terminates. A prime power chosen once never occurs again. The total deleted mass is consequently less than

$$\frac{\eta\xi}{2\log\log N}\sum_{q\le N}\frac1q <\eta\xi$$

for sufficiently large $N$. In this sum $q$ ranges over prime powers; Mertens' estimate gives $\sum_{q\le N}1/q=\log\log N+O(1)$. The terminal set therefore has the required mass, and termination means every remaining fiber satisfies the desired lower bound.

Dependencies and verification

The prime-power Mertens estimate follows from the prime estimate recorded as an external input in [[unit_fractions/liu_2024_further_questions_regarding_unit_fractions/theorem_2_1|Theorem 2.1]] and the convergence of $\sum_p1/(p(p-1))$. The pruning method is the same as [[unit_fractions/bloom_2021_density_conjecture_about_unit_fractions/lemma_6|Bloom's Lemma 6]]. This rewritten proof passed independent blind review on 2026-09-18, retained as the fresh main-proof review with its distinct grade; the earlier main-proof review was ruled on 2026-09-18 a coordinated compilation check, not an independent review.

Bears on

• Problem 298
• Problem 299
• Problem 300
• Problem 310