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The Wiener Wintner Theorem Along the Primes

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves a Wiener–Wintner theorem along the primes, with return-times convergence for every second system.

desk verdict A serious, novel extension of return-times/Wiener–Wintner ergodic theorems along the primes, but the current manuscript has load-bearing gaps (Lemma 2.13's unresolved case and Lemma 5.1's omitted proof) that block acceptance as-is. read the letter →

arxiv 2601.10459 v3 pith:MPK6MTRC submitted 2026-01-15 math.DS math.CAmath.NT

Jan Fornal , Anastasios Fragkos , Ben Krause , Michael Lacey , Hamed Mousavi , Yu-Chen Sun This is my paper · ORCID
classification math.DSmath.CAmath.NT MSC 37A3037A4511N0511B30
keywords Wiener-WintnertheoremReturntimesprimenumbersvonMangoldtfunctionGowersnormsHeath-BrownmodelpointwiseergodichigherorderFourieranalysis
topics The Twin Prime Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's goal is to prove the Return Times Theorem with averages taken over the prime numbers, and the Wiener–Wintner theorem as a corollary. It claims that for any measure-preserving system and any bounded function f, there is a full-measure set of starting points x such that for every second system (Y,S) and every bounded g, the averages (1/N)∑_{n≤N} f(T^{p_n}x)g(S^{p_n}y) converge almost everywhere in y. For the Wiener–Wintner part, the same convergence holds for all weights e^{2πi p_n θ} and for f in L^p with p>1. This matters because it is the first extension of these classical pointwise ergodic theorems to a sparse arithmetic sequence, and it requires blending the Heath–Brown decomposition of the von Mangoldt function with higher-order Fourier analysis and uniform Wiener–Wintner estimates. The full text states the main theorem as a Return Times Theorem; the abstract emphasizes the Wiener–Wintner corollary.

What carries the argument

The central object is the Heath–Brown model Λ_Q(n)=∑_{Q/2<q≤Q} μ(q)/φ(q)c_q(n), a rational-complexity approximation to the von Mangoldt function. The authors prove a fixed-complexity U^3 bound ∥Λ_Q∥_{U^3[N]} ≲ Q^{-3/8+o(1)} — an improvement beyond the easy Q^{-1/4} bound that is essential to the argument — and a U^3 approximation of the full von Mangoldt function by the model truncated at Q=exp((log N)^{1/10}). These bounds feed a transference argument in which U^3-small weights are plugged into Gowers–Cauchy–Schwarz inequalities, reducing the prime averages to uniform Wiener–Wintner estimates on weakly mixing functions.

What would settle it

Give a concrete configuration of 5 marked vertices on the cube {0,1}^3 with no face containing 4 marked vertices for which no single green vertex, run through the described algorithm, colours all 5 vertices green; that would falsify Lemma 2.13 and remove the Q^{-3/8} bound. Alternatively, compute directly the U^3 norm of Λ_Q for large Q and find a lower bound exceeding Q^{-3/8+o(1)}.

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Extended reading notes

Core claim

The central claim is Theorem 1.4: along the sequence of primes p_n, the return-times averages converge pointwise for all second systems, and the Wiener–Wintner averages converge for each fixed frequency θ. The mechanism is to write the prime indicator through the von Mangoldt function, approximate it by a truncated Heath–Brown model, control the U^3 norm of the part of the model with fixed rational complexity Q by the bound Q^{-3/8+o(1)}, and then transfer Gowers-norm control of the weight to almost-everywhere convergence of weighted ergodic averages via uniform Wiener–Wintner and weak-mixing reductions.

Load-bearing premise

The proof stands on the combinatorial counting lemma that bounds the number of solutions to the integrality system derived from the cube {0,1}^3, and the text reports that the case f(5)=1 of that lemma is still unresolved; if that counting bound fails, the Q^{-3/8} U^3 estimate collapses, and with it the proof of the main theorems.

Editorial extensions

If this is right

  • For any bounded function on any measure-preserving system, the prime Wiener–Wintner averages converge for every frequency on a full-measure set; the L^p version holds for p>1.
  • The return-times statement holds: for a full-measure set of starting points, the sequence f(T^{p_n}x) is a universally good weight for pointwise L^2 ergodic theorems along the primes.
  • The fixed-complexity U^3 estimate gives quantitative control on how close the Heath–Brown model is to a prime-weight sequence, making it reusable in other prime-weighted ergodic averaging problems.
  • The argument shows that a weight with sufficiently small U^3 norm is enough to force almost-everywhere convergence in both the Wiener–Wintner and the return-times form, not just L^2 convergence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the fixed-complexity U^3 estimate generalizes to U^s norms — the paper states such a generalization is forthcoming — the same strategy may yield analogues for averages over primes in polynomial or multicorrelation settings.
  • The unresolved f(5)=1 case of Lemma 2.13 is the most checkable point of the proof; verifying or refuting it decides whether the Q^{-3/8} improvement, and with it the present proof of both main theorems, stands.
  • The method suggests a testable extension: apply the same Heath–Brown-plus-U^3 decomposition to other sparse sequences such as primes plus squares, for which direct computation of the relevant U^3 norm could be compared against the conjectured bounds.
  • A reading that separates the two stated results is useful: the Wiener–Wintner corollary is presented as depending on the same U^3 machinery but with a weaker uniformity requirement, so a purely Wiener–Wintner proof might survive even if the full return-times statement needs adjustment.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper claims two results: a Wiener–Wintner theorem along the prime times (for f∈L^p, p>1, with a full-measure set of x making the exponential prime averages converge for every frequency), and a Return Times theorem along the primes (for f,g∈L∞, convergence for all second systems). The strategy is to approximate the von Mangoldt function by a Heath–Brown model, prove a fixed-complexity U^3 bound (Proposition 2.1) and a U^3 approximation theorem (Proposition 3.1), and then transfer uniform Wiener–Wintner/return-times information via Gowers-norm inequalities (Proposition 1.14) and transference arguments (Lemmas 4.1 and 5.1). The central claimed contribution is the U^3 control of Heath–Brown models.

Significance. If correct, the results would be significant: they give the first Wiener–Wintner and Return Times theorems along an arithmetic sequence, namely the primes. The proof architecture is credible and non-circular: it imports the integer Wiener–Wintner/return-times theorems, the uniform Wiener–Wintner input, Wierdl's prime ergodic theorem, and quantitative U^3 inverse theorems, with no fitted parameters. The fixed-complexity U^3 estimate for Heath–Brown models is a plausible independent contribution. However, the manuscript currently leaves several load-bearing proofs unfinished (Lemma 2.13, Lemma 3.23, Lemma 5.1) and contains a false periodicity bound in the proof of Lemma 4.1. I am not convinced the central claims are established as written.

major comments (4)
  1. [Section 2, proof of Lemma 2.13] Lemma 2.13 is the combinatorial engine behind inequality (2.9), which in turn yields the Q^{-3/8} bound in Proposition 2.1; the introduction states that improvement beyond Q^{-1/4} is essential. The proof asserts f(1)=f(2)=f(3)=0 and f(4)=1, but for |S|=5 it contains the sentence 'The case with f(5)=1 is still unresolved' and then refers to Figure 2 instead of giving the case analysis. Since |S|=5 requires |T|=1, an unresolved case invalidates (2.9). Also, with T empty, the stated algorithm cannot color a one-vertex S, so f(1)=0 is not justified as stated. This is a load-bearing gap, not a presentation issue.
  2. [Section 4, Lemma 4.1 (period bound for Λ_Q)] The proof asserts that Λ_Q(n) is periodic with period P_Q := lcm(q≈Q) and 'recall the bounds 2Q ≤ P_Q ≤ 3Q'. This is false: for Q=8, the relevant squarefree q in (4,8] are 5,6,7, whose lcm is 210, while 3Q=24; in general the lcm of the dyadic interval is far larger than Q. The bound P_Q ≤ 3Q is used to make P_Q/N0 small in the first case (3Q≤K); without it, the block decomposition of L^θ_{Q,N} does not establish the claimed estimate. Lemma 4.1 therefore has a concrete error at a load-bearing point.
  3. [Section 5, Lemma 5.1] Lemma 5.1 is the return-times analogue of Lemma 4.1 and the central transference estimate for the prime Return Times theorem. The proof is omitted with the phrase 'very close to that of Lemma 4.1, and so we omit the details.' Lemma 4.1's proof is long and relies on specific U^3 and periodicity estimates; the extra dependence on g_x(y-kn) and L^2(Y) norms is not a routine modification. This omission prevents verification of Section 5.
  4. [Section 3, Lemma 3.23] Lemma 3.23 provides the required bound on products of Ramanujan sums (3.24) and is used in the proof of Lemma 3.19 to control the Siegel-zero correction (3.18). Its proof is reduced to a counting argument 'very similar to Lemma 2.13' and then left to the reader. Given that Lemma 2.13 is itself incomplete, this deferred proof is especially problematic. Proposition 3.1 is not fully demonstrated without it.
minor comments (5)
  1. [Title/abstract] The abstract in the paper text highlights the Return Times theorem while the arXiv metadata highlights the Wiener–Wintner theorem; make these consistent.
  2. [Section 2, proof of Proposition 2.1] In the final parameterization, the condition 's|r, s|r^4' should be 'rad(s)|r, s|r^4' (and the surrounding text should be adjusted accordingly).
  3. [References] References [14] and [15] are the same book with inconsistent bibliographic data; they should be merged or distinguished clearly.
  4. [Remark 1.18] Remark 1.18 contains an unproved inequality with 'we omit the details'; since the remark is not used in the main proof, either move it to future work or provide the proof.
  5. [Notation] The notation X≪Y is introduced with an 'extremely large' implicit constant, which is nonstandard; use the standard Vinogradov convention or define it precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the prime Wiener–Wintner / Return Times derivation uses independent ingredients; the flagged unresolved combinatorial case is a correctness gap, not a circular step.

full rationale

I find no step in the derivation that reduces to its own input by construction. Theorem 1.4 is proved from independent ingredients: the classical integer Wiener–Wintner and Return Times theorems, Wierdl's prime ergodic theorem, Bourgain's uniform Wiener–Wintner estimate, and U^3 inverse theorems of Leng/Sanders/Green–Tao. The Heath–Brown model Λ_Q is an explicit formula, not defined in terms of the prime-weighted averages being proved, and Proposition 2.1 is a separate quantitative U^3-counting statement. Proposition 3.1 uses Vaughan decomposition and an independent arithmetic-progressions estimate (Lemma A.1), not the target theorem. The self-citations that supply auxiliary moment bounds ([17, Lemma 4.6], [17, (4.7)]) are not the claimed Wiener–Wintner/Return Times conclusions and so do not create a circular chain. The manuscript does contain a serious internal gap: Lemma 2.13's proof says 'The case with f(5)=1 is still unresolved' and the key bound (2.9) is the exclusive source of the Q^{-3/8} improvement, which the introduction calls essential ('the weaker bound on the U^3 norm of at most Q^{-1/4} is easily accessible. Yet, for our proof an improvement beyond 1/4 is essential'). Lemma 5.1 is also deferred ('The proof of this Lemma is very close to that of Lemma 4.1, and so we omit the details'). These are correctness/completeness risks, not circularity: failure of these lemmas would break the proof, but it would not make the proof's conclusion equal to its assumptions by construction. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No data-fitting free parameters appear. The central claim rests on deep external inverse-theorem machinery (Leng preprints), on standard analytic number theory inputs, and on three internal lemmas whose proofs are incomplete or deferred. These are treated as axioms of the current version of the paper.

assumptions (7)
  • domain assumption Quantitative inverse U^3 theorem (Theorem 3.7) and nilsequence equidistribution statements from Leng [19,20], including [20, Prop 5.1, Lemmas B.5/B.6, Lemma A.6].
    Used in Lemma 3.5 to convert a U^3 lower bound for Λ−Λ'_{≤Q} into correlation with a degree-2 nilsequence; these are arXiv preprints not proved in the paper.
  • ad hoc to paper Lemma 2.13: for every marked-vertex set S⊂{0,1}^3 there is a small T⊂S such that the green-blue algorithm terminates; yields the counting bound (2.9).
    New combinatorial assertion unique to this paper; proof includes the 'f(5)=1 is still unresolved' phrase and relies on figure-based case analysis; it supplies the essential Q^{-3/8} estimate.
  • ad hoc to paper Lemma 3.23 bound on the U^3 expectation of products of Ramanujan sums.
    Reduction is sketched and details are left to the reader; needed for the Siegel-zero correction estimate in Theorem 3.1.
  • ad hoc to paper Lemma 5.1: the return-times analogue of Lemma 4.1, the transference estimate for prime averages.
    Proof is omitted ('very close to Lemma 4.1'); this lemma is essential for the final Return Times Theorem conclusion.
  • standard math Siegel zero existence/uniqueness, Landau–Page theorem, and prime number theorem in arithmetic progressions (e.g., [14, Ch. 5], [15, Thm 5.27]).
    Used to define Λ'_{≤Q} and to control Λ−Λ'_{≤Q} on progressions (Lemma A.1).
  • standard math Uniform Wiener–Wintner bound for weakly mixing f (1.8) and the classical Return Times Theorem [3,4,6,7].
    Used as black-box inputs in the transference arguments; they are independent of the prime-weighted versions being proved.
  • standard math Gowers–Cauchy–Schwarz inequality and Proposition 1.14 inequalities (1.15)–(1.17).
    Core inequalities transferring U^3 control of the weight to ergodic averages; these are proved in the text.

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Cite this review

Pith. "Pith review of The Wiener Wintner Theorem Along the Primes." pith.science (2026). https://pith.science/paper/MPK6MTRC

@misc{pith2026260110459,
  author       = {Pith},
  title        = {Pith review of: The Wiener Wintner Theorem Along the Primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPK6MTRC}},
  note         = {Machine review of arXiv:2601.10459}
}
abstract

We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, \nu),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $\nu(X_f) = 1,$ so that for all $\omega \in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 \pi i p_n \theta} f(T^{p_n} \omega) \] converges for all $\theta \in [0,1]$; above, $\{2 = p_1 < p_2 < \dots\}$ are an enumeration of the primes. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with U^3 theory playing an important role; our $U^3$-estimates for Heath-Brown models of the von Mangoldt function may be of independent interest.

Figures

Figures reproduced from arXiv: 2601.10459 by the authors.

Figure 1
Figure 1. An example of labelling of a cube with green and blue marked vertices in the case |S| = 6. (1) Query whether there exists a face with all marked vertices but one vertex being green. If that is not the case, then we quit the algorithm. (2) If the algorithm runs, then the equation (2.11) determines the value aωp. (3) That said, we color the vertex ω green and we return to the initial query. Lemma 2.13. For any collect… view at source ↗
Figure 2
Figure 2. The four cases, up to symmetry, in which there are 5 marked vertices on the cube, with 1 being green, and no face having 4 marked ver￾tices. f(5) = 1 is still unresolved, but now we can without loss of generality assume that bottom face contains 3 vertices in S and top face contains 2 vertices in S. Up to rotation with respect to the top-bottom axis, there are only 4 possible alignments for which the condition with … view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Variation Norm Carleson Theorem Along the Primes

    math.CA 2026-07 accept novelty 7.5 of 10

    The variational prime Carleson operator is bounded on ℓ^p for an r-dependent range c(r)<p<C(r) that approaches the full (1,∞) as r o∞, and the maximal version is bounded for all 1<p<∞.

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