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Statement

For a set $E$ of positive integers, write $R(E)=\sum_{n\in E}1/n$, $E_d=\{n\in E:d\mid n\}$, and $\mathcal Q_E=\{p^a:p^a\mid n\text{ for some }n\in E,\ a\ge1\}$. Brackets denote least common multiples, and $Q=[\mathcal Q_A]=\operatorname{lcm}(A)$. An integer is $S$-smooth in this paper when every prime-power divisor, not just every prime divisor, is at most $S$. Put $e(u)=\exp(2\pi iu)$.

There is an absolute constant $C\ge1$ with the following property. Fix $\delta\in(0,1)$ and $\varepsilon\in(0,1/10)$, and take $N$ sufficiently large in terms of these two parameters. Suppose $\eta\ge(\log N)^{-1}$ and

$$N^{0.99}\le S\le K\le M\le N/10^4.$$

Define

$$\Gamma=\max\left\{ \frac{\eta}{(\log N)^\delta(\log\log N)^3}, \frac{\eta^2(\log N)^{1-2\delta}} {\log^2(N/M)(\log\log N)^5}\right\}.$$

Assume also

$$S\le\min\left\{\frac{M^2}{CN},\frac{\eta MK^2}{N^2(\log N)^3}\right\}, \qquad K\le M\exp(- (\log N)^{1-\delta}),$$

and

$$C\le\frac{\Gamma^2} {(\log N)^{2\varepsilon}(\log(N/M)+(\log N)^{1-\delta})}.$$

Let $A\subseteq[M,N]$ satisfy

• every $n\in A$ is $S$-smooth;
• $\Omega(n)\le5\log\log N$ for every $n\in A$;
• $\min_{q\in\mathcal Q_A}qR(A_q)\ge\eta$.

If $x$ is a positive integer and

$$(1+(\log N)^{-1})\frac xQ\le R(A)\le(\log N)\frac xQ,$$

there is $B\subseteq A$ such that $R(B)=x/Q$.

Source: Liu–Sawhney, arXiv:2404.07113v1, Proposition 5.2, printed/PDF pp. 16–19 (the statement is on p. 16 and the last proof estimate is on p. 19). The source uses the first term in $\Gamma$ for Theorem 1.1 and the second for Theorem 1.3 and Proposition 1.4.

Rewritten proof

The proof follows pp. 16–19 with explicit corrections to the residue notation, Fourier threshold, dyadic endpoints and parameter estimates. It uses the proved application form of Lemma 5.1, rather than its false unrestricted statement as printed. The corrections are recorded below; none is attributed to an author erratum or to the uninspected published version.

Feasible parameters and the divisor scale

Write

$$L=\log N,\quad\ell=\log\log N,\quad w=\log(N/M), \quad a=L^{1-\delta}.$$

The hypotheses give $\log10^4\le w\le.01L$. The positive mass condition makes $A$ nonempty. For any $q\in\mathcal Q_A$, we have $q\le S\le Me^{-a}$, so harmonic summation gives

$$\eta\le qR(A_q) \le\sum_{M/q\le m\le N/q}\frac1m \le w+O(q/M)\le2w.$$

Let $\Gamma_1,\Gamma_2$ denote the two terms in $\Gamma$. If $\delta\ge1/2$, then

$$\frac{\Gamma_1^2}{L^{2\varepsilon}(w+a)} \le\frac{4w}{L^{2\delta+2\varepsilon}\ell^6}\to0, \qquad \frac{\Gamma_2^2}{L^{2\varepsilon}(w+a)} \le\frac{16L^{2-4\delta-2\varepsilon}}{\ell^{10}(w+a)}\to0.$$

This contradicts the last hypothesis. Thus every feasible fixed choice has $\delta<1/2$. In particular,

$$\Gamma\gg L^\varepsilon\sqrt a =L^{\varepsilon+(1-\delta)/2}\gg L^{3\varepsilon},$$

since $\varepsilon<1/10$. The divisor lemma will be applied with mass parameter $\eta/2$. Its scale is therefore

$$H_*=e^\rho,\qquad\rho=\frac{\eta a}{2\ell^3w},\qquad \Gamma=\max\left\{\frac{2\rho w}{L}, \frac{4\rho^2\ell}{L}\right\}.$$

The final hypothesis forces $\Gamma\to\infty$. A bounded subsequence of $\rho$ would make both expressions on the right bounded, so $\rho\to\infty$. Thus $H_*\ge2$ for large $N$, as required by the corrected application form of Lemma 5.1.

Fourier reduction and concentration

Set

$$\tau=\frac{x/Q}{R(A)},\qquad \frac1{\log N}\le\tau\le\frac{\log N}{1+\log N}.$$

Include each $n\in A$ independently in $B$ with probability $\tau$. Integer Fourier orthogonality gives

$$\begin{aligned} \mathbb P(R(B)-x/Q\in\mathbb Z) &=\sum_{B\subseteq A}\tau^{|B|}(1-\tau)^{|A\setminus B|} \frac1Q\sum_{-Q/2<h\le Q/2} e\!\left(\sum_{n\in B}\frac hn-\frac{hx}Q\right)\\ &=\frac1Q\sum_{-Q/2<h\le Q/2} e(-hx/Q)\prod_{n\in A}(1-\tau+\tau e(h/n))\\ &=\frac1Q\sum_{-Q/2<h\le Q/2} \operatorname{Re}\!\left(e(-hx/Q) \prod_{n\in A}(1-\tau+\tau e(h/n))\right). \end{aligned} \tag{5.1–5.2}$$

The last equality also follows by conjugate symmetry. It will suffice to show that this probability is at least $1/(2Q)$ and that the probability of a nonzero integer discrepancy is less than $1/(4Q)$. Since $\mathbb ER(B)=x/Q$ and each summand changes by at most $1/M$, Lemma 2.6 gives

$$\mathbb P(|R(B)-x/Q|\ge1) \le2\exp(-M^2/(2N)).$$

By smoothness and Theorem 2.1, $Q\le\prod_{q\le S}q\ll e^{5S}$. The condition $S\le M^2/(CN)$, with $C$ large, implies $2\exp(-M^2/(2N))\le e^{-6S}<1/(4Q)$ for sufficiently large $N$.

Major arcs

We first justify the cardinality needed by Lemma 3.1. Put $y_0=e^{a/(10\ell)}$, and let $p_0$ be the smallest prime dividing an element of $A$. If $p_0\ge y_0$, each $n/p_0$ with $n\in A_{p_0}$ avoids every prime below $y_0$. Furthermore $M/p_0\ge M/S\ge e^a$. Lemma 2.4 applied to doubling intervals, followed by reciprocal summation across $O(w)$ such intervals, gives

$$\eta\le p_0R(A_{p_0})\ll\frac w{\log y_0} =\frac{10w\ell}{a}.$$

The sieve cutoff holds because the local logarithmic scale is at least $a+O(1)$ and its second logarithm is at most $\ell+o(1)$. The displayed bound contradicts $\eta=2\rho\ell^3w/a$ and $\rho\to\infty$. Hence $p_0<y_0$, and

$$|A|\ge M R(A_{p_0})\ge\frac{\eta M}{p_0} \ge\frac M{Ly_0}\ge N^{.99-o(1)}>N^{.95}.$$

We may therefore apply Lemma 3.1 with all inclusion probabilities equal to $\tau$, obtaining

$$\frac1Q\sum_{|h|\le M/2} \operatorname{Re}\!\left(e(-hx/Q) \prod_{n\in A}(1-\tau+\tau e(h/n))\right)\ge\frac3{4Q}.$$

The allowed interval for $\tau$ is inside $[(\log N)^{-2},1-(\log N)^{-2}]$ for large $N$, as required by Lemma 3.1.

Minor arcs: decay

For each $n$, let $h_n\in(-n/2,n/2]$ be the integer congruent to $h$ modulo $n$, and put

$$t=\frac{50N^2L\ell}{\tau(1-\tau)K^2},\qquad I_h=(h-K/2,h+K/2).$$

Define the exceptional prime powers by

$$\mathcal D_h=\left\{q\in\mathcal Q_A: |\{n\in A_q:|h_n|\ge K/2\}|<t\right\}.$$

Each $n$ belongs to at most $\Omega(n)\le5\log\log N$ of the sets $A_q$. Applying Fact 2.5 gives

$$\begin{aligned} \left(\prod_{n\in A}|1-\tau+\tau e(h/n)|\right)^{5\log\log N} &\le\prod_{q\in\mathcal Q_A}\prod_{n\in A_q} |1-\tau+\tau e(h/n)|\\ &\le\prod_{q\in\mathcal Q_A\setminus\mathcal D_h} \exp(-2\tau(1-\tau)K^2t/N^2)\\ &\le\exp(-100|\mathcal Q_A\setminus\mathcal D_h| \log N\log\log N). \end{aligned}$$

Taking the $1/(5\ell)$ power gives $N^{-20}$ per exceptional complement element, and in particular

$$\prod_{n\in A}|1-\tau+\tau e(h/n)| \le N^{-10|\mathcal Q_A\setminus\mathcal D_h|}. \tag{5.3}$$

For each large-residue factor we used $|1-\tau+\tau e(h/n)|\le\exp(-2\tau(1-\tau)K^2/N^2)$.

Minor arcs: a common multiple near h

For $q\in\mathcal D_h$ let

$$T_q=\{n\in A_q:|h_n|<K/2\}.$$

With the definition of $\mathcal D_h$, $|A_q\setminus T_q|<t$ and

$$R(T_q)\ge R(A_q)-t/M\ge\eta/q-t/M\ge\eta/(2q).$$

Indeed, $\tau(1-\tau)\ge1/(2L)$ for the allowed probability range, so the stated bound on $S$ gives

$$\frac tM\le\frac{100N^2L^2\ell}{MK^2} \le\frac\eta{2S}$$

for large $N$, since $L\ge200\ell$ eventually. Apply the corrected application form of Lemma 5.1 to $T_q$ with mass parameter $\eta/2$, obtaining $d_q$ and $T_q^*\subseteq(T_q)_{qd_q}$. Its global prime-factor bound is inherited from $A$; we already proved $\delta<1/2$ and $H_*\ge2$. The size condition for $q$ follows from $q\le S\le K\le M\exp(- (\log N)^{1-\delta})$. It gives

$$qd_q\ge M\exp(- (\log N)^{1-\delta})\ge K.$$

There is at most one multiple of $qd_q$ in $I_h$. Since a nonempty $T_q^*$ consists of integers with a multiple in $I_h$, there is such a multiple; call it $x_q$. Put

$$\widetilde T_q=\{n/(qd_q):n\in T_q^*\}.$$

The reciprocal-mass conclusion gives

$$R(\widetilde T_q)=qd_qR(T_q^*) \ge\frac\eta{C(\log N)^\delta\log\log N}.$$

The same output gives $\min\widetilde T_q\ge H_*$ and $\max\widetilde T_q/\min\widetilde T_q\le e^w$. Put $E=\widetilde T_q$ and use all dyadic bins $[2^j,2^{j+1})$ with

$$\lfloor\log_2\min E\rfloor\le j\le\lfloor\log_2\max E\rfloor.$$

This includes the bin containing the minimum. Because $\rho\to\infty$, the indices are large and positive. Their reciprocal weights satisfy

$$W:=\sum_j\frac1{j+1} \ll\min\left\{\ell,\frac{w+1}{\rho}\right\} \ll\min\left\{\ell,\frac{2w^2\ell^3}{\eta a}\right\}.$$

The first estimate follows either by summing $1/(j+1)$ up to $O(L)$ or by bounding the number of bins by $O(w+1)$ and the smallest index below by a constant times $\rho$. The second uses $w\ge\log10^4$. Since the bins partition $E$,

$$R(E)\le W\max_j\bigl((j+1)R(E\cap[2^j,2^{j+1}))\bigr).$$

Thus some $y_q=2^j$ has

$$\begin{aligned} R(E\cap[y_q,2y_q)) &\ge\frac c{\log y_q} \max\left\{\frac\eta{L^\delta\ell^2}, \frac{\eta^2a}{w^2L^\delta\ell^4}\right\}\\ &=\frac{c\ell\Gamma}{\log y_q} \ge\frac\Gamma{\log y_q} \end{aligned}$$

for large $N$. The spare factor $\ell$ absorbs fixed constants. The selected bin base obeys $\min E/2\le y_q\le\max E$, so in particular $y_q\le N/(qd_q)$. Its reciprocal mass is at most a fixed constant; hence $\log y_q\gg\Gamma\gg L^{3\varepsilon}$.

Now fix $q_1,q_2\in\mathcal D_h$, set $y=\min(y_{q_1},y_{q_2})$, and define the following set of primes:

$$\mathcal P=\{p:\exp((\log N)^\varepsilon)\le p \le\exp((\log y)(\log N)^{-\varepsilon})\}.$$

The preceding lower bound on $\log y$ makes this interval nonempty. Its upper prime logarithm is at most $(\log y_q)L^{-\varepsilon}$ for either bin. As $L^{-\varepsilon}\le1/\sqrt{\log\log y_q}$ eventually, Lemma 2.4 applies to both bins. For $q=q_1,q_2$, let

$$\mathcal P_q=\{p\in\mathcal P:p\nmid m \text{ for every }m\in\widetilde T_q\cap[y_q,2y_q)\}.$$

Applying Lemma 2.4 to $[y_q,2y_q)$, all selected integers avoid the primes in $\mathcal P_q$, giving

$$R(\mathcal P_q)\le-\log\!\left(\frac\Gamma{C_s\log y_q}\right).$$

Here $C_s$ is an absolute sieve comparison constant, independent of the statement constant $C$. The sieve gives an upper bound proportional to $\prod_{p\in\mathcal P_q}(1-1/p)$ for the reciprocal mass, which is at most a constant times $\exp(-R(\mathcal P_q))$. Combining this with the proved lower bound gives the displayed logarithm. By the reciprocal-prime estimate in Theorem 2.1,

$$\begin{aligned} R(\mathcal P\setminus(\mathcal P_{q_1}\cup\mathcal P_{q_2})) &\ge R(\mathcal P)-R(\mathcal P_{q_1})-R(\mathcal P_{q_2})\\ &\ge\log\frac{\log y}{2(\log N)^{2\varepsilon}} +\log\frac\Gamma{C_s\log y_{q_1}} +\log\frac\Gamma{C_s\log y_{q_2}}\\ &=\log\frac{\Gamma^2} {2C_s^2(\log N)^{2\varepsilon}\log\max(y_{q_1},y_{q_2})}\\ &\ge\log\frac{\Gamma^2} {2C_s^2(\log N)^{2\varepsilon}(\log(N/M)+(\log N)^{1-\delta})} \ge1. \end{aligned}$$

The penultimate inequality uses $y_q\le N/(qd_q)$ and the lower bound on $qd_q$. The final inequality uses the proposition's last hypothesis, choosing its absolute constant $C\ge2eC_s^2$.

For $p\in\mathcal P\setminus\mathcal P_q$, some $n\in T_q^*$ has $p\mid n/(qd_q)$. Since $|h_n|<K/2$, the integer $h-h_n\in I_h$ is a multiple of $n$, hence a multiple of $pqd_q$. Uniqueness of the multiple of $qd_q$ in $I_h$ gives $h-h_n=x_q$, and therefore $p\mid x_q$. Every prime in $\mathcal P\setminus(\mathcal P_{q_1}\cup\mathcal P_{q_2})$ divides $x_{q_1}-x_{q_2}$. Their reciprocal sum is at least 1 and each is at least $u=\exp((\log N)^\varepsilon)$, so there are at least $u$ such primes and their product is at least $u^u>N$. But $|x_{q_1}-x_{q_2}|<K\le N$. Thus $x_{q_1}=x_{q_2}$. There is a single integer $z\in I_h$ divisible by every element of $\mathcal D_h$, and hence by $[\mathcal D_h]$. The source calls this integer $x$, overloading the target numerator; $z$ separates those roles here.

Summing the minor arcs

Fix $D\subseteq\mathcal Q_A$. If $\mathcal D_h=D$, a multiple of $[D]$ lies in $I_h$. Among a complete period of $Q$ values of $h$, the number of such $h$ is bounded by

$$(K+1)\frac{[\mathcal Q_A]}{[D]} \le N\prod_{q\in\mathcal Q_A\setminus D}q \le N^{|\mathcal Q_A\setminus D|+1}. \tag{5.4}$$

For $|h|>M/2\ge K/2$ in the centered period, $\mathcal D_h\ne\mathcal Q_A$: otherwise a multiple of $Q$ would lie within $K/2$ of such $h$, which is impossible. Combining (5.3) and (5.4), and using at most $N^s$ choices of a complement of size $s$, yields

$$\begin{aligned} \frac1Q\sum_{D\subsetneq\mathcal Q_A} N^{|\mathcal Q_A\setminus D|+1}N^{-10|\mathcal Q_A\setminus D|} &\le\frac1Q\sum_{s\ge1}N^{s+1}N^sN^{-10s}\\ &\le\frac2{QN}. \end{aligned}$$

The major arcs therefore contribute at least $3/(4Q)$ and the minor arcs have absolute contribution at most $2/(QN)$. For large $N$, (5.2) is at least $1/(2Q)$. Subtracting the concentration bound leaves positive probability that $R(B)=x/Q$, as required.

Source corrections and verification

The source's p. 17 residue definition omits the absolute value, and $T_q$ is printed as a cardinality rather than a set. The proof above uses the meanings required by the ensuing argument. Its Fourier threshold retains $(1-\tau)^{-1}$; the existing $L^3$ slack in the bound on $S$ absorbs that factor. The source's assertion $R(A)\ge\eta$ is replaced by the smallest-prime sieve argument that gives the needed major-arc cardinality.

The actual Lemma 5.1 scale has mass parameter $\eta/2$ and denominator $\ell^3$. The printed pp. 17–18 instead mix powers $\ell^2$ and $\ell$; the proof keeps the actual $H_*$ and includes the initial dyadic bin. Weighted pigeonholing retains the claimed $\Gamma$, with a spare factor $\ell$ to absorb constants. The feasibility estimates explicitly give $\delta<1/2$, $H_*\to\infty$, and $\Gamma\gg L^{3\varepsilon}$, closing its invocation and prime-interval checks. These bounded corrections received independent review before incorporation (the source checks and preliminary review). The exact rewritten proof also 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. No assertion is made about changes in the uninspected published version.

Dependencies

• Theorem 2.1: bounds the common denominator and the reciprocal mass of the prime interval.
• Lemma 2.4: bounds the exceptional primes avoided by the selected dyadic set.
• Fact 2.5: the modulus estimate for each Fourier factor.
• Lemma 2.6: concentration around the target reciprocal sum.
• Lemma 3.1: positive contribution from major arcs.
• Lemma 5.1: selects a divisor with reciprocal mass and a lower bound on scaled elements in its proved application form.

The source credits Croot [7] and Bloom [4, Propositions 2 and 3] for the proof framework. The elementary Fourier orthogonality argument is included above and is not an additional black-box dependency on Proposition 3.2.

Bears on

• Problem 298, through Theorem 1.1.
• Problem 299, through Theorem 1.1.
• Erdős problem 300, through Theorem 1.3.
• Erdős problem 310, through Proposition 1.4.