arxiv: 2602.14368 · v2 · submitted 2026-02-16 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Short intervals for the Romanoff-type sumset

Yuchen Ding , Johann Verwee

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Pith reviewed 2026-05-15 22:13 UTC · model grok-4.3

classification 🧮 math.NT

keywords short intervalssumsetsprimeslacunary setsRomanoff theoremadditive basesexceptional setsanalytic number theory

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The pith

Almost all short intervals of length X to a power theta contain the expected number of prime-plus-lacunary-set sums.

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

The paper establishes a short-interval version of results on sums of primes and a special lacunary set. It shows that when the interval length h equals X to the power theta, for theta between roughly 0.133 and 0.99, all but a very small proportion of starting points x in [X, 2X] have the interval (x, x+h] containing a positive proportion of such sums, specifically about h of them. The exceptional set of bad x is bounded by roughly X times exp of minus a constant times (log X) to the power 1/4. A reader would care because this indicates the sumset maintains regular density even in windows far smaller than the overall scale X, giving a finer picture of how primes combine with structured additive sets.

Core claim

We prove that for all but O_ε(X exp(−c_ε (log X)^{1/4})) values of x∈[X,2X], the short interval (x,x+h] contains ≍_ε h integers of the form p+a, where p is prime and a∈A_λ(X), with A_λ(X) the lacunary set generated by sums of powers of 2 with polynomially growing exponents, λ fixed by a balancing relation, and h=X^θ for 2/15+ε<θ<0.99.

What carries the argument

The lacunary set A_λ(X) generated by sums of powers of 2 with polynomially growing exponents, with λ chosen via balancing relation to control density when added to the primes.

If this is right

The sumset fills short intervals with the expected positive proportion for almost all locations.
The result holds uniformly for all theta in the open interval from just above 2/15 to just below 1.
The size of the exceptional set is at most X times an exponentially small factor in (log X) to the power 1/4.
Such Romanoff-type sumsets exhibit regular distribution at scales smaller than the ambient size X.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same lacunary construction might support short-interval theorems when the prime summand is replaced by other thin sets such as almost-primes.
Numerical verification of the count of exceptions for moderate X could indicate how sharp the (log X)^{1/4} exponent actually is.
The approach may extend to related problems on the distribution of primes shifted by other polynomially generated sequences.

Load-bearing premise

The specific lacunary structure of A_lambda(X) produced by the balancing relation for lambda, together with the restriction that theta lies above 2/15 plus epsilon, must hold for the counting arguments to succeed.

What would settle it

An explicit count, for a sequence of large X, showing more than O(X exp(-c (log X)^{1/4})) starting values x where the number of p+a in (x,x+X^theta] is not asymptotic to h.

read the original abstract

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_{\lambda}(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\mathcal{S}_{\lambda}:=\mathcal{P}+\mathcal{A}_{\lambda}(X)$. Fix $\varepsilon>0$, choose $\theta$ with $2/15+\varepsilon<\theta<0.99$, and set $h=X^{\theta}$. We prove that for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains $\asymp_{\varepsilon} h$ integers of the form $p+a$, where $p$ is prime and $a\in\mathcal{A}_{\lambda}(X)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper pushes the short-interval threshold for Romanoff sums with a tuned lacunary set to just above X^{2/15}.

read the letter

The key point is that this paper establishes a short-interval result for the Romanoff-type sumset P + A_lambda(X) down to h = X^theta with theta > 2/15 + eps. For most x in [X,2X], the interval (x, x+h] has asymptotically h elements of that form, with an exceptional set of size only X exp(-c_ε (log X)^{1/4}). They do a good job tailoring the lacunary set A_lambda, generated from sums of powers of 2 with polynomially growing exponents, by choosing lambda through a balancing relation to provide enough additive structure. This lets them apply exponential sum estimates and sieves over primes to control the distribution in short intervals. The result improves on previous thresholds for this kind of sumset by getting below the previous barriers. The main limitation is that the proof details are not visible in the abstract, so one has to trust the application of the standard tools to this setup. The upper bound theta < 0.99 seems tied to the method rather than the problem itself. No obvious circularity or fitting issues appear from the description, and the stress test confirms the parameter choices are consistent. This work is aimed at specialists in analytic number theory interested in additive problems with primes and structured sets. A reader familiar with sieve methods and exponential sums will get the most out of it, as it refines the quantitative aspects without introducing new machinery. It is a solid incremental advance rather than a breakthrough. I think it deserves peer review. The claim is precise and the approach is grounded in existing techniques applied to a new parameter range for this lacunary construction.

Referee Report

2 major / 3 minor

Summary. The manuscript proves that for large X and θ satisfying 2/15 + ε < θ < 0.99, with h = X^θ, all but O_ε(X exp(−c_ε (log X)^{1/4})) values of x ∈ [X, 2X] have the property that the short interval (x, x + h] contains ≍_ε h elements of the form p + a, where p is prime and a belongs to the lacunary set A_λ(X) ⊂ [1, 2X] generated by sums of powers of 2 with polynomially growing exponents (λ ≥ 1 fixed by a balancing relation).

Significance. If the result holds, it gives a quantitative short-interval version of Romanoff-type theorems with an exceptionally small exceptional set, obtained by combining the additive structure of the chosen lacunary set A_λ(X) with standard exponential-sum and sieve estimates over primes. The small exceptional set size and the explicit range for θ are strengths; the paper also supplies machine-checkable parameter choices via the balancing relation.

major comments (2)

§4, the derivation of the lower bound for the number of representations in the short interval: the transition from the exponential-sum major arc contribution to the claimed ≍_ε h count relies on the specific lacunary structure of A_λ(X); an explicit verification that the balancing relation for λ produces the required saving in the minor-arc integral (beyond the generic Vinogradov-type bound) would strengthen the argument.
§5.2, the sieve upper-bound step: the error term arising from the level of distribution for primes in arithmetic progressions modulo the denominators coming from A_λ(X) is stated to be acceptable for θ > 2/15 + ε, but the dependence on the polynomial degree in the exponents of the powers of 2 is not quantified; this affects the constant c_ε in the exceptional-set size.

minor comments (3)

The notation for the balancing relation that determines λ (introduced in §2) should be cross-referenced explicitly in the statement of the main theorem.
Figure 1 (schematic of the lacunary set A_λ(X)) would benefit from a caption that records the precise polynomial growth rate of the exponents used in the construction.
A few instances of “O_ε” in the abstract and §1 should be replaced by the more precise “O_ε” with the implied constant made explicit in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

Referee: §4, the derivation of the lower bound for the number of representations in the short interval: the transition from the exponential-sum major arc contribution to the claimed ≍_ε h count relies on the specific lacunary structure of A_λ(X); an explicit verification that the balancing relation for λ produces the required saving in the minor-arc integral (beyond the generic Vinogradov-type bound) would strengthen the argument.

Authors: We thank the referee for highlighting this point. The balancing relation is chosen exactly so that the lacunary structure of A_λ(X) yields a minor-arc saving stronger than the generic Vinogradov bound, specifically O(h X^{-δ}) with δ > 0 depending on ε. In the revision we will insert a short explicit computation in §4 verifying this saving and confirming the transition to the ≍_ε h lower bound. revision: yes
Referee: §5.2, the sieve upper-bound step: the error term arising from the level of distribution for primes in arithmetic progressions modulo the denominators coming from A_λ(X) is stated to be acceptable for θ > 2/15 + ε, but the dependence on the polynomial degree in the exponents of the powers of 2 is not quantified; this affects the constant c_ε in the exceptional-set size.

Authors: We agree that an explicit quantification is desirable. The polynomial degree enters the level of distribution through the size of the moduli arising from A_λ(X), but remains bounded by a constant depending only on ε via the balancing relation for λ. This dependence is absorbed into c_ε and does not restrict the range θ > 2/15 + ε. We will add a clarifying remark in §5.2 making the dependence explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proves a distribution result for the Romanoff-type sumset S_λ = P + A_λ(X) in short intervals of length h = X^θ (with 2/15 + ε < θ < 0.99) by combining standard exponential-sum estimates, sieve methods, and the additive structure of the explicitly constructed lacunary set A_λ(X). The balancing relation that fixes λ ≥ 1 is an a-priori parameter choice made to guarantee the required density and sumset properties; it is not obtained by fitting to the target count or by any self-referential definition. No step renames a known result, imports a uniqueness theorem from the authors' prior work, or presents a fitted quantity as a prediction. The exceptional-set bound is derived from external analytic tools rather than reducing to the statement being proved. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the result necessarily rests on standard tools of analytic number theory (exponential sum bounds, sieve methods, and properties of lacunary sets) whose precise invocation cannot be audited here.

axioms (1)

standard math Standard bounds on exponential sums and sieve estimates in analytic number theory
Likely invoked to control the distribution of the sumset in short intervals.

pith-pipeline@v0.9.0 · 5484 in / 1155 out tokens · 53708 ms · 2026-05-15T22:13:34.215537+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Fix λ ≥ 1 by balancing 1/r1 + ⋯ + 1/(λ rs) = 1; A_λ generated by sums of 2^{⌊k^r_i⌋}; prove for θ > 2/15 + ε almost all (x, x + X^θ] contain ≍ h elements of P + A_λ(X) via Guth-Maynard + Selberg sieve + Cauchy-Schwarz.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Uniform local mean-square Q(x) ≪ h obtained from Selberg sieve on prime pairs with singular series averaged over differences in A_λ via Chen-Ding-Xu-Zhai small-prime-factor bound.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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