REVIEW
Products of two elements of Granville's random model in residue classes
Number Theory · Submitted 29 Sep 2026 · 26 references
Pith's verdict: The mathematics is sound; the only genuine defect is a one-line overstatement in the Corollary 4.2 proof, and the model-comparison with Banks–Ford–Tao is the genuinely interesting part.
T0 · AI review
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Referee report
The paper
The paper claims
Every reduced residue class modulo a large $q$ contains a product of two elements of Granville's random model of the primes lying below $q^{\Theta}$, for any fixed $\Theta > 3/4$, almost surely, and the number of such products is…
The reading
What carries the argument
The load-bearing object is Granville's discard-then-keep construction. On each dyadic block $(X, 2X]$, every integer $n$ with a prime factor at most $A(X) = \log X \cdot e^{-\sqrt{\log\log X}}$ is discarded deterministically; each survivor is kept independently with probability $P_n = (Q_X/\varphi(Q_X))/\log n$, where $Q_X$ is the product of the primes $\leq A(X)$. Determinism supplies the small-modulus bias of the primes, so the analogue holds for every reduced class, while independence makes the representation count $N_q(a)$ a sum of jointly independent Bernoulli variables indexed by the orbits of the fixed-point-free involution $n \mapsto \bar{a}n$ on $(X, 2X]$. The counting input is Lemma 3.1, the classical estimate for lattice points on the modular hyperbola $m_1 m_2 \equiv b \pmod{q}$: the count is $\frac{\varphi(q)}{q^2}|J_1||J_2| + O(\tau(q)^2 q^{1/2}\log^2 q)$, obtained by additive-character detection, completion, and the Weil–Estermann bound on Kloosterman sums. The balance between that error and the main term fixes the $\Theta > 3/4$ threshold; Bennett's inequality and Borel–Cantelli then turn the precise no-representation probability and the concentration estimate into the two almost-sure statements.
Extended reading notes
Core claim
The central claim is Theorem 1.1. Fix $\Theta \in (3/4, 1]$ and a choice of the parameter $A$ in Granville's model; for each modulus $q \geq 16$ let $X_q$ be the largest power of two with $2X_q \leq q^{\Theta}$. Then, almost surely, there is a number $q_0$ such that for every modulus $q \geq q_0$ and every reduced class $a \in (\mathbb{Z}/q\mathbb{Z})^{\times}$, there exist distinct $g_1, g_2 \in G \cap (X_q, 2X_q]$ with $g_1 g_2 \equiv a \pmod{q}$. Moreover, writing $N_q(a)$ for the number of such ordered pairs, one has almost surely $\max_a \bigl| N_q(a)\cdot q\log^2 X_q /(c_q X_q^2) - 1\bigr| \to 0$, with $c_q = \frac{\varphi(q)}{q}\prod_{p \leq A(X_q),\, p \mid q}(1 - p^{-1})^{-2}$ the natural local correction. Since $(X_q, 2X_q] \subseteq [1, q^{\Theta}]$, this is the analogue of Erdős–Odlyzko–Sárközy in the wider range $q^{\Theta}$, for all moduli, and the exponent is sharp for the method: the error $O(q^{1/2+o(1)})$ from the lattice-point count balances the main term $X^2/q$ when $X = q^{3/4}$, and the statement certainly fails below $\Theta = 1/2$.
Load-bearing premise
The theorem is an almost-sure statement on Granville's probability space, and the space itself does the work: the conclusion transfers to the genuine primes only if Granville's discard-then-keep model is accepted as a faithful picture of them, since a differently constructed random model of the primes fails the same statement with positive probability.
Editorial extensions
If this is right
- The analogue of the EOS conjecture holds unconditionally for Granville's model in the range $q^{\Theta}$ for any fixed $\Theta > 3/4$ and all moduli, whereas for genuine primes the conjecture at the range $q$ remains open even under the GRH.
- The representation count is asymptotically $c_q X_q^2/(q\log^2 X_q)$ with $c_q$ the natural local factor, so the model reproduces not only representability but the expected accounting for every reduced class.
- The exponent $3/4$ is the sharp limit of this method, fixed by the deterministic balance between the modular-hyperbola main term and the error $O(q^{1/2+o(1)})$; below $\Theta = 1/2$ the statement certainly fails.
- Cramér's model (the case $T = 1$) also works, with $c_q$ degenerating to $\varphi(q)/q$, but it does not see the small-modulus bias of the primes; Granville's deterministic discard is what gives the model both that bias and independence.
- Model choice is decisive: in the Banks–Ford–Tao sieve model the analogue fails with positive probability when $\gcd(q,6) \neq 1$, because random deletion of one residue class per small prime leaves every retained element even (or divisible by 3) with probability $1/2$.
Reading between the lines
- The paper fixes the method's ceiling at $3/4$, not the phenomenon's: any saving on incomplete Kloosterman sums would push the valid range below $3/4$, and the true critical exponent inside Granville's model is left open between $1/2$ and $3/4$.
- Because the probabilistic content is minimal, the orbit-and-hyperbola scheme appears to transfer to any set with the same independence-after-discard structure; an analogous argument would likely prove the corresponding statement for $k$-fold products of model elements, with the threshold shifting as the main term becomes $X^k/q$.
- Read as a diagnostic, the contrast identifies the operative question for real primes: whether their small-modulus bias behaves like Granville's deterministic discard, which preserves independence, or like a sieve model's random deletion of residue classes, which breaks it, since only the former structure appears to support the EOS conclusion.
Original abstract
Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range $q^{\Theta}$ for any fixed $\Theta > 3/4$, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range $q^{\Theta}$ for any fixed $\Theta > 3/4$, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. | background 0.89 | 0.02 | 0.03 | speculative 0.61 |
| s1 | We prove that its analogue holds, unconditionally and in the wider range $q^{\Theta}$ for any fixed $\Theta > 3/4$, in Granville's random model of the primes. | result 0.82 | 0.96 | 0.89 | asserted 1.00 |
| s2 | Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. | result 1.00 | 0.90 | 0.49 | asserted 0.99 |
| s3 | The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes. | interpretation 0.53 | 0.70 | 0.06 | asserted 0.99 |
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References
Discussion
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