REVIEW 3 major objections 3 minor 4 references
Additive Problems with Primes from a Thin Bohr Set
T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read The paper claims that for any irrational $\alpha$ and $\tau<1/8$, the set of primes $p$ with $\lVert\alpha p\rVert\le p^{-\tau}$ contains infinitely many nontrivial three-term arithmetic progressions.
desk verdict The abstract is a plausible new result about primes in thin Bohr sets, but the supplied full text is an unrelated lozenge-tiling paper, so there is nothing to referee. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The operative object is the thin Bohr set $\mathcal{P}_\tau(\alpha)$, defined by the distance condition $\lVert\alpha p\rVert\le p^{-\tau}$; the threshold $\tau<1/8$ is the range in which the paper claims the required analytic control holds. The argument turns on distribution estimates for primes in these short Bohr neighborhoods—how uniformly the primes spread over intervals of multiples of $\alpha$—since that distribution is what lets one count three-term progressions inside the selected set. The binary Goldbach variant is carried by the same distribution mechanism.
What would settle it
For a fixed irrational $\alpha$ with known continued fraction expansion, enumerate primes $p\le X$ and count triples $(p_1,p_2,p_3)$ in arithmetic progression with $p_1<p_2<p_3\le X$ and $\lVert\alpha p_i\rVert\le p_i^{-\tau}$ for a fixed $\tau\in(0,1/8)$. If this count does not grow without bound as $X$ increases—or vanishes for some $\tau<1/8$—the infinitude claim is false; comparing the count's order with the claimed lower bound would test the proof mechanism directly.
Extended reading notes
Core claim
The central claim is that for an irrational $\alpha$ and fixed $\tau\in(0,1/8)$, the set $\mathcal{P}_\tau(\alpha)=\{p\text{ prime}:\lVert\alpha p\rVert\le p^{-\tau}\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of length three still occur infinitely often. The paper further states a binary Goldbach-type result for sums of two primes from the same thin Bohr set.
Load-bearing premise
The theorem rests on a nontrivial analytic estimate controlling how many primes fall into the thin Bohr neighborhoods of multiples of $\alpha$; if that estimate is assumed rather than proved, the infinitude result is conditional.
Editorial extensions
If this is right
- For every irrational $\alpha$, prime three-term progressions exist even after restricting to primes within $p^{-\tau}$ of a multiple of $\alpha$, for every $\tau<1/8$.
- The same thin selection is compatible with additive representations: the announced binary Goldbach-type result would give infinitely many binary representations using two primes from the same thin Bohr set.
- The exponent range $\tau\in(0,1/8)$ provides a quantitative benchmark for how sparse a Diophantine selection can be before additive structure disappears.
- If the underlying distribution estimate is proved for the stated range, it supplies a new counting input for additive problems over density-zero prime subsets defined by rotations.
Reading between the lines
- One testable extension is to compute, for a concrete irrational $\alpha$ such as $\sqrt{2}$, the number of three-term progressions up to $X$ in $\mathcal{P}_\tau(\alpha)$ and compare its growth with the order implied by the claimed proof; a collapse for some $\tau<1/8$ would indicate a hidden dependence on the Diophantine quality of $\alpha$.
- The threshold $1/8$ may come from the available exponential-sum bounds rather than from a structural barrier; a plausible stronger statement would be that the same infinitude holds for all $\tau$ below the irrationality-measure exponent of $\alpha$, though the paper does not claim this.
- The body text supplied with this submission concerns nonintersecting paths, Pfaffians, and lozenge tilings, and contains none of the prime-counting argument; the summary above follows the abstract. If the body is the intended manuscript, the abstract's claims are not supported by the visible text.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The abstract of arXiv:2508.12139 claims that for a fixed irrational alpha and for tau in (0, 1/8), there exist infinitely many nontrivial three-term arithmetic progressions consisting of primes p satisfying ||alpha p|| <= 1/p^tau. A binary Goldbach-type variant is also announced. The supplied full text, however, is not the corresponding number theory paper: it is arXiv:2508.12135, a combinatorics paper on a reflection principle for nonintersecting paths and lozenge tilings with free boundaries. It contains no statement about primes, Bohr sets, exponential sums, or the parameter tau. Thus, as submitted, the manuscript's central claim is unsupported by any proof or even by a theorem statement beyond the abstract.
Significance. If the abstract's claim were proved unconditionally, it would be a noteworthy additive number theory result: it would show additive structure inside a thin Bohr-type subset of the primes for arbitrary fixed irrational alpha, with a nontrivial exponent range tau < 1/8, together with a binary Goldbach-type analogue. Such a result would likely require a delicate exponential-sum or level-of-distribution estimate for primes restricted by ||alpha p|| <= p^{-tau}, and the threshold 1/8 would be a substantive quantitative feature. However, the manuscript as supplied gives the referee no way to assess the derivation: the full text is an unrelated combinatorics paper, and the abstract contains no estimates, no hypotheses beyond alpha irrational and tau in (0,1/8), and no indication whether the claim is unconditional or conditional on an unproved hypothesis. The potential significance cannot currently be credited.
major comments (3)
- [Full text] The body of the submission is not the paper described by the abstract. It is a lozenge-tiling and nonintersecting-paths paper (arXiv:2508.12135) with no mention of primes, Bohr sets, the condition ||alpha p|| <= 1/p^tau, or additive problems. The central claim of the abstract therefore has no proof in the submitted manuscript. This is a load-bearing omission, not a presentational issue: the claimed theorem is exactly as strong as the missing analytic estimate, and no such estimate is stated or proved anywhere in the supplied text.
- [Abstract] Even treating the abstract as the only mathematical content, it omits the analytic input necessary to make the claim checkable. A result of this type requires a nontrivial bound on exponential sums over primes weighted by a Bohr-neighborhood condition of vanishing width p^{-tau}, or an equivalent level-of-distribution statement. The abstract does not state such an estimate, does not say whether it is proved or assumed, and does not indicate whether the theorem is unconditional or conditional on, for example, GRH or a conjecture. Consequently the claimed exponent range tau < 1/8 cannot be verified from the manuscript as submitted.
- [Abstract] The announced binary Goldbach-type result is not formulated. It is unclear whether the statement is an analogue for sums of two primes satisfying the Bohr condition, whether it is an asymptotic or an infinitude statement, and what error terms are claimed. Without a precise statement and proof, this part of the abstract is not assessable.
minor comments (3)
- [Abstract] The phrase 'non-trivial three-term arithmetic progressions' is not defined; presumably it means progressions with distinct primes, but this should be stated.
- [General] The manuscript gives no references to prior work on primes in Bohr neighborhoods, Diophantine approximation with primes, or related exponential-sum estimates. Such context would be needed in a complete submission.
- [Full text] The full text appears to be a different article with its own title, abstract, and numbering. This mismatch should be resolved editorially; the current submission is not suitable for normal peer review as it stands.
Circularity Check
No circularity found; the supplied full text is a different paper and contains none of the abstract's claimed derivation.
full rationale
No circular step can be identified because the supplied full text (arXiv:2508.12135, Byun, on lozenge tilings) does not contain the derivation behind the abstract's claim about primes in thin Bohr sets. Circularity requires exhibiting a specific reduction in which a claimed prediction is equivalent to its own input by construction, a fitted parameter is renamed as a prediction, or a load-bearing conclusion is justified solely by a self-citation chain. The lozenge-tiling manuscript's arguments use independent external results (Lindström–Gessel–Viennot, Okada–Stembridge, Okada's Pfaffian formula, and independently proved product formulas by Hopkins–Lai and Okada) and do not define their conclusions in terms of their inputs. The abstract/full-text mismatch is a serious verification and provenance concern, but it is not a circularity; therefore the appropriate circularity score is 0.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Additive Problems with Primes from a Thin Bohr Set." pith.science (2026). https://pith.science/paper/BZTMQVWY
@misc{pith2026250812139,
author = {Pith},
title = {Pith review of: Additive Problems with Primes from a Thin Bohr Set},
year = {2026},
howpublished = {\url{https://pith.science/paper/BZTMQVWY}},
note = {Machine review of arXiv:2508.12139}
}
abstract
For an irrational $\alpha\in \mathbb{R}$, we consider additive problems with the set of primes satisfying $\lVert\alpha p\rVert\leq \frac{1}{p^\tau}$ for some fixed $\tau>0$. In particular, we show that there exist infinitely many non-trivial three-term arithmetic progressions in the set of primes satisfying $\lVert \alpha p\rVert\leq \frac{1}{p^\tau}$ for $\tau\in(0, \tfrac18)$. We also consider a binary Goldbach-type problem.
Reference graph
Works this paper leans on
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work page Pith review arXiv 2025
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plane partitions and \symmet- ric and self-complementary
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Reviewed August 5, 2026 · model on record in the stance chip above.
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