REVIEW 3 major objections 4 minor 3 references
Popular differences in primes along fractional powers
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read For non-integer c>2, the primes contain infinitely many pairs separated by floor(m^c).
desk verdict A genuinely new result on prime pairs with fractional-power differences, but the proof currently has a repairable log-power gap in Lemma 3.5 that should be fixed before publication. 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 proof uses Fourier analysis on $\mathbb{Z}/N\mathbb{Z}$, rewriting the double average as a character sum of $|\widehat{\Lambda}(\xi)|^2$ against an exponential sum in $\lfloor m^c\rfloor$. The trivial character contributes the main term, and the nontrivial characters are split into a 'close' range, $\xi\le N/\log^b N$, and a 'far' range, $\xi>N/\log^b N$. For the close range, Hölder's inequality separates the Fourier mass of $\Lambda$ from a 2k-th moment of the exponential sum, and that moment is bounded by counting solutions to the Diophantine inequality $|\sum m_i^c - \sum m_j^c|\le k$. For the far range, the maximum of the exponential sum is factored out, the floor function is replaced by $m^c$ through a partition of the circle, a discrepancy inequality controls the replacement, and classical exponential-sum bounds for smooth phases finish the estimate.
What would settle it
Re-derive Proposition 3.2 while keeping the $\log^4 N$ factor from Theorem 2.6 visible: Lemma 3.5's bound becomes $O(N\log^4 N/\log^{b/2}N)$, and with $b=4kA+2Bc+4k$ the Hölder step yields only $O(\log^{-(A+(Bc-4)/k)}N)$. If this computation is carried through, the claimed $O(\log^{-A}N)$ conclusion is not established, and the proof would need a sharper estimate or a different split.
Extended reading notes
Core claim
The central claim is Theorem 1.1: for any non-integer c>2 and any A≥1, with B=(A+2)/c and M=$N^{{1/c}}$/\log^B N, the average $E_{m\le M}E_{n\le N} \Lambda(n)\Lambda(n+\lfloor m^c\rfloor)$ equals $1+O(\log^{-A}N)$. The main term 1 is exactly what one would get if $\Lambda(n)$ and $\Lambda(n+\lfloor m^c\rfloor)$ behaved independently, and the error shrinks faster than any fixed power of $1/\log N$. A direct consequence is Corollary 1.2: infinitely many pairs (p,m) with p and p+⌊m^c⌋ both prime, and in fact about $N M \log^{-2}N$ such pairs with p≤N and m≤M. This is the fractional-power analogue of the known polynomial-pattern asymptotics for the primes.
Load-bearing premise
The load-bearing premise is that the uniform bound on the Fourier transform of the von Mangoldt function over the close characters holds with the full logarithmic savings stated in Lemma 3.5; if the extra $\log^4 N$ factor from the cited Fourier estimate is not absorbed, the final $O(\log^{-A}N)$ error term does not follow as written.
Editorial extensions
If this is right
- For every non-integer c>2, the primes contain infinitely many pairs (p,m) with p and p+⌊m^c⌋ both prime.
- The number of such pairs with p≤N and m≤M is approximately $N M \log^{-2}N$, matching the heuristic independence count.
- The double average is $1+o(1)$, the same leading constant as for polynomial patterns, so fractional powers introduce no extra bias at this order.
- For any fixed A, taking $M=N^{1/c}/\log^{(A+2)/c}N$ makes the error $O(\log^{-A}N)$; increasing A improves the error only by shortening the m-range.
Reading between the lines
- Editorial extension: The same Fourier-decoupling scheme should transfer to multiple fractional-power differences, for instance differences of the form $\lfloor m^{c_1}\rfloor+\lfloor m^{c_2}\rfloor$, if a fractional analogue of the quoted Diophantine bound exists; this is the direction the paper's Conjecture 4.2 points toward.
- Editorial extension: Numerical evaluation of the double average for c=2.5 or c=3.5 at moderate N would expose whether the logarithmic error is real and how large the implied constant is; the paper reports no such checks.
- Editorial extension: If the close-character estimate loses a few powers of log, a slightly weaker theorem with A replaced by $A-(Bc-4)/k$ may still hold, because the Hölder decomposition absorbs logarithmic losses without changing the structure.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Cesàro average E_{m≤M} E_{n≤N} Λ(n)Λ(n+⌊m^c⌋) for non-integer c>2 and M=N^{1/c}/log^B N. Theorem 1.1 claims this average equals 1+O(log^{-A}N) for any A≥1 when B=(A+2)/c, and Corollary 1.2 concludes that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence {⌊m^c⌋}. The proof rewrites the average via the discrete Fourier transform on Z/NZ (Proposition 2.10), decomposes the resulting character sum into a main term, a 'close to trivial' part Σ1, and a 'far' part Σ2, and estimates Σ1 using Parseval, Vinogradov's bound for the Fourier transform of Λ, and Poulias's Diophantine inequality, and Σ2 using the Erdős-Turán inequality and van der Corput estimates.
Significance. If the proof is completed, the result is a natural complement to the Tao–Ziegler theorems on polynomial patterns in the primes: it gives the first asymptotic for prime pairs with difference in a non-integer power sequence, and the frequency matches the naive independence heuristic. The paper is clearly structured, gives a dependency graph, and states precise conjectures for generalizations. The main term is benchmarked against the Prime Number Theorem, and the argument is built from external standard estimates rather than assuming the conclusion; there is no evidence of circularity or fitted constants. The identified gaps are localized and appear repairable, so the work has the potential to be a publishable contribution after revision.
major comments (3)
- [Section 3.1, Lemma 3.5] The proof of Lemma 3.5 applies Theorem 2.6 to obtain |Λhat(ξ)| ≤ C(N/√q + N^{4/5} + √(qN)) log^4 N. Since q > log^{b/2}N, the first term is at most N log^4 N / log^{b/4}N, that is, N/log^{b/4-4}N, not N/log^{b/4}N as written. The displayed estimate after applying Vinogradov therefore does not yield the stated bound O(N/log^{b/4}N). With the paper's choice b=4kA+2Bc+4k, the Hölder step in Proposition 3.2 produces O(log^{-A+4/k}N), which is weaker than the claimed O(log^{-A}N). This is load-bearing because Lemma 3.5 controls the entire contribution of Σ1. The flaw is localized and repairable by enlarging b (for instance b=4kA+2Bc+4k+16), but the proof as written does not deliver the theorem.
- [Section 3, Eq. (3.1)] The displayed identity (3.1) is false as written. The left-hand side sums over ξ=0,...,N-1, while the right-hand side is the main term plus Σ1 over 1≤ξ≤N/log^b N plus Σ2 over N/log^b N≤ξ≤N/2, omitting the range N/2<ξ≤N-1. The sentence 'Since |Λhat(ξ)|=|Λhat(-ξ)|, we can reduce to ξ≤N/2' is not an equality; the omitted terms are the conjugates of the corresponding half-range terms and must be bounded separately or incorporated via 2Re. The final paragraph of Lemma 3.15 invokes symmetry, but Eq. (3.1) itself is incorrect. The decomposition should be rewritten with the symmetric contribution made explicit.
- [Section 3.2, Lemmas 3.9 and 3.10] The definitions of µ1, µ2 and c1, c2 use cos(-2πi ξm^c/N) and sin(-2πi ξm^c/N). With the imaginary unit inside, these are hyperbolic functions, not bounded by 1, and the claim that µ1 and µ2 are probability measures is false. The product-to-sum identities in Lemma 3.10 show that the intended arguments are the real quantities cos(2πξm^c/N) and sin(2πξm^c/N). Without correcting these definitions, Lemma 3.10 and hence Proposition 3.14 (and therefore Proposition 3.3) are not established. Additionally, the assertion |c1|,|c2|≤1/2 in Lemma 3.10 is not proved; the proof uses 1/(1+c1)≤2, so a uniform lower bound for 1+c1 is needed.
minor comments (4)
- [Section 3.1, Lemma 3.6] The symbols r and ξ are both used for the frequency, and k is re-used for ⌊log^m N⌋ in conflict with the global k in Theorem 2.8; this makes the proof harder to follow.
- [Section 3.2, Proposition 3.14] The phrase 'noting that j = 2 j+1 ≈ M' is garbled; it should say that the largest dyadic term occurs at j ≈ log_2 M.
- [Section 2.4, Proposition 2.10] The text says 'apply Fourier inversion to each of the indicator functions', but the objects being transformed are the von Mangoldt function, not indicators; this is a wording issue.
- [Section 3.1, Lemma 3.7] The proof should state explicitly that N is taken large enough so that kN/log^{Bc}N < N, since this is needed for the geometric sum evaluation to reduce to the R=0 case.
Circularity Check
No circularity: the main term is benchmarked to the Prime Number Theorem, all inputs are external standard results, and auxiliary parameters are freely chosen; the flagged Lemma 3.5 log^4 gap is a repairable technical error, not a circular reduction.
full rationale
FINDING: no significant circularity. Theorem 1.1 is derived from external, independently established inputs: the main term 1 is benchmarked to the Prime Number Theorem (Theorem 2.5), the close-character sum Sigma_1 is controlled through Vinogradov's bound on the Fourier transform of Lambda (Theorem 2.6), Parseval's identity, Holder's inequality, and Poulias's bound on solutions of a fractional Diophantine inequality (Theorem 2.8, an external 2021 result), while the far-character sum Sigma_2 is handled via the Erdos-Turan inequality (Theorem 2.7), van der Corput exponential-sum bounds (Theorem 2.9, Graham-Kolesnik), and a dyadic decomposition. There are no fitted inputs: the auxiliary parameters B, b, k and the frequency cut-off log^b N are freely chosen functions of the target accuracy A, and the claimed error O(log^{-A} N) is nowhere assumed; it emerges from the sizes of the external bounds. The heuristic that floor(m^c) destroys multiplicative structure is presented only as motivation ('it is natural to conjecture...') and plays no logical role in the proof. No reference is authored by the present five authors, so there is no self-citation chain to carry any load-bearing premise. The reviewer-flagged issue in Lemma 3.5 - Vinogradov's log^4 N factor is not absorbed, so the printed derivation gives only N/log^{b/4-4}N and hence a weaker exponent in Proposition 3.2 - is a genuine but localized and repairable arithmetic gap (one would enlarge b by about 16k), not a circular step: Lemma 3.5's bound still comes from an external theorem, and the target O(log^{-A}N) is not an input to any lemma. The paper's own limitation statement (the restriction c > 2 is needed because 'a lemma used in [Pou21] breaks down') concerns a technical hypothesis, not a circularity. Verdict: no significant circularity, score 0.
Assumptions & free parameters
assumptions (6)
- standard math Prime number theorem in the explicit form of Theorem 2.5 (Montgomery-Vaughan).
- standard math Vinogradov's Fourier estimate for the von Mangoldt function (Theorem 2.6).
- standard math Poulias's Diophantine inequality bound (Theorem 2.8).
- standard math van der Corput exponential sum bound (Theorem 2.9).
- standard math Erdős-Turán discrepancy inequality (Theorem 2.7).
- standard math Dirichlet approximation theorem.
Cite this review
Pith. "Pith review of Popular differences in primes along fractional powers." pith.science (2026). https://pith.science/paper/M72U5ZNV
@misc{pith2026241117599,
author = {Pith},
title = {Pith review of: Popular differences in primes along fractional powers},
year = {2026},
howpublished = {\url{https://pith.science/paper/M72U5ZNV}},
note = {Machine review of arXiv:2411.17599}
}
abstract
We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c} \log^{-B} N$. As a combinatorial consequence, we obtain that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence $\bigl\{\lfloor m^c \rfloor \colon m \in \mathbb{N} \bigr\}$ for any non-integer $c > 2$.
Figures
Reference graph
Works this paper leans on
-
[1]
A new upper bound for sets with no square differences
[BM22] Thomas F. Bloom and James Maynard. “A new upper bound for sets with no square differences”. In: Compositio Mathematica 158.8 (2022), pp. 1777–1798. doi: 10.1112/S0010437X22007679. [Dav67] Harold Davenport. Multiplicative Number Theory. 2nd ed. Graduate Texts in Mathematics. Springer,
-
[1967]
On the probability that n and [nc] are coprime
Chap. 25, pp. 143–144. isbn: 978-1-4757-5927-3. doi: 10.1007/978-1-4757-5927-3 . [DD02] Francine Delmer and Jean-Marc Deshouillers. “On the probability that n and [nc] are coprime”. In: Periodica Mathematica Hungarica 45 (2002), pp. 15–20. [Des73] Jean-Marc Deshouillers. “Probl`eme de Waring avec exposants non entiers”. In: Bulletin de la Soci´et´e Math´e...
-
[2006]
Diophantine inequalities of fractional degree
isbn: 978-0-521-84903-6. [Pou21] Constantinos Poulias. “Diophantine inequalities of fractional degree”. In:Math- ematika 67.4 (2021), pp. 949–980. doi: 10.1112/mtk.12112. [PSS88] J´anos Pintz, W. L. Steiger, and Endre Szemer´edi. “On sets of natural numbers whose difference set contains no squares”. In: Journal of the London Mathemat- ical Society s2–37.2...
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.