A comment on the number of $k$-th powers inside arithmetic progressions

Sa\v{s}a Novakovi\'c

REVIEW 2 major objections 5 minor 90 references

A comment on the number of $k$-th powers inside arithmetic progressions

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Sharp count of k-th powers survives polynomial step growth

desk verdict A small, honest corollary extension of Bourgain–Demeter with a sloppy write-up: the math likely holds, but the induction mislabeling and abstract/Theorem mismatch need fixing. read the letter →

arxiv 2607.15895 v1 pith:NDM4JEQW submitted 2026-07-17 math.NT

Saša Novaković This is my paper · ORCID

classification math.NT MSC 11B2511D4511N25

keywords k-thpowersarithmeticprogressionsdivisorfunctionsharpupperboundspolynomialvaluesestimatessub-polynomialgrowthnumbertheory

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 shows that a known sharp upper bound for the number of k-th powers inside an arithmetic progression—originally proved only when the progression's step has at most O(1) divisors—continues to hold, up to an N^ε factor, when the step q grows at most polynomially with the length N. The argument is short: it combines the prior bound for lower-degree polynomials with a standard estimate on the divisor function of q. A weaker intermediate bound (N^{2/k}) is obtained under the milder condition d(q) ≤ N^{1/(k(k+1))} log^{O(1)}. The paper openly notes that both results were likely already known.

What carries the argument

The divisor-counting function d(q) and its maximal growth are the engine. The known sharp bound is of the form d(q)^k N^{1/(k+1)} up to constants; the paper pairs it with a classical theorem stating that for q ≤ cN^r, d(q) is at most N^{o(1)}, so d(q)^k ≤ N^ε for large N. The proof's factorization of t−t0 and P_k(t) into complementary pieces is the other load-bearing mechanism, but the divisor estimates do the actual work of turning the prior theorem into the new uniform statement.

What would settle it

A concrete falsifier would be to find a single k and an infinite family of triples (a,q,N) with q≤N^r such that a degree-k polynomial takes more than N^{1/k+ε} values in {a+q,...,a+Nq}, for arbitrarily large N. Numerically, searching k=2, q≈N^r and a chosen to maximize square density would provide evidence; if the count ever exceeds N^{1/2+ε}, Theorem 1.4 is false.

Watch

Extended reading notes

Core claim

The central claim is that the factor d(q)^k appearing in the known bound can be absorbed into N^ε whenever d(q) is sub-polynomial in N, so the near-optimal N^{1/k+ε} count holds uniformly for every step q ≤ cN^r and every sufficiently large N. The paper also records an intermediate theorem: if d(q) ≲ N^{1/(k(k+1))} log^{O(1)}, the count is ≲ N^{2/k}. The proofs are brief inductions that factor a difference of two polynomial values into complementary divisors of q; a classical estimate on the maximal size of d(q) then converts the divisor factor into a negligible loss.

Load-bearing premise

The proof depends entirely on the prior sharp bound [3] for lower-degree polynomials: if that bound fails, or has hidden restrictions on the step, both theorems collapse; the paper's induction does not prove the base bound, it imports it.

Editorial extensions

If this is right

For any fixed r, all arithmetic progressions with step q ≤ N^r contain at most N^{1/k+ε} k-th power values (for N sufficiently large), uniformly in the starting value a.
This nearly matches the conjectured optimal N^{1/k} order, closing the gap up to N^ε for the entire polynomial-step range.
The intermediate N^{2/k} bound under d(q) ≤ N^{1/(k(k+1))} log^{O(1)} gives a nontrivial bound for steps with moderately many divisors, not just O(1).
The result applies not only to sequences a+qx but to values of any integer-coefficient degree-k polynomial, so it covers non-linear progressions as well.
Since the proof is a direct combination of the prior sharp bound with standard divisor estimates, the hard part of the problem is already contained in the base theorem.

Reading between the lines

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

The paper's induction, as written, never invokes the theorem being proved; in the lower-degree step it uses the prior sharp bound [3] directly. This means the two theorems are one-line corollaries of [3] plus divisor estimates, not genuinely new inductive results.
The critical parameter for uniform N^{1/k+ε} is the maximal order of d(q) along q ≤ cN^r; any theorem bounding this maximum by N^{o(1)} would yield the same conclusion, so the result is robust to the specific divisor estimate used.
For k=2, the condition in Theorem 1.2 becomes d(q) ≲ N^{1/6}; since d(q) for q ≤ N^r is usually much smaller, the new N^{2/k} bound is crude but the method suggests that the N^{1/2+ε} bound for squares might hold under far weaker restrictions than polynomial steps.
A testable extension: replace d(q) by other multiplicative functions to see if analogous counts for values of polynomial forms remain near-optimal; the factorization step would carry over if the function satisfies a similar sub-polynomial growth.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know: this is a tiny comment, not a research paper. The two theorems are basically unwrapped corollaries of Bourgain–Demeter plus Wigert's divisor bound. The author says so himself, which is to his credit.

What's actually new: Theorem 1.4 gives a uniform N^{1/k+ε} bound for all q ≤ N^r, for every fixed degree k, once N is large. That's not literally in Bourgain–Demeter or in Hajdu–Papp, and it's a useful observation. Theorem 1.2 relaxes the divisor-count condition to d(q) ≤ N^{1/(k(k+1))} polylog and gets N^{2/k}. Both are easy, as the author admits.

Where it gets soft: the proof is arranged as an induction, but the 'induction hypothesis' is actually the external Bourgain–Demeter theorem. The author never proves or restates that base bound, and the step labelled 'due to the induction hypothesis' is a direct application of BD to a shorter progression. That's fine as a proof, but the mislabeling makes the paper look self-contained when it isn't. A referee should ask for a rewrite that isolates exactly which theorem is being invoked.

More worrying is the inconsistency between the abstract and Theorem 1.1. The abstract says BD only handled steps with O(1) divisors; Theorem 1.1 states BD's bound with d(q)^{k-1} for all q. If the abstract is right, the proofs have no valid base. If Theorem 1.1 is right, the abstract is wrong. I suspect the BD theorem actually has the divisor factor, since the reader's report recovered the inequalities from it, but this mismatch must be fixed before the paper can be trusted. The author needs to check the citation and phrase it accurately.

Minor: 'sharp' is used without lower bounds; there's no discussion of matching lower constructions for these uniform bounds. And the author's own admission that the results 'should be known' invites a literature check.

Who is this for: someone writing in the area of powers in arithmetic progressions who wants a quick uniform bound with polynomial q. It's not deep, but it's usable.

In peer review, I'd send it out rather than desk-reject: the claims are concrete and the central issue—whether BD's theorem is quoted correctly—is checkable. I'd expect the referee to ask for a revision that fixes the induction language and the abstract mismatch. If the BD citation checks out, it's publishable as a short note; if not, it's not.

Referee Report

2 major / 5 minor

Summary. The paper makes two observations about the number of k-th powers in an arithmetic progression. Theorem 1.2 states that if the divisor count of the step satisfies d(q) ≲ N^{1/(k(k+1))}, then any degree-k integer polynomial takes values in the progression at most N^{2/k+o(1)} times. Theorem 1.4 states that if q ≲ N^r, then for every ε>0 and N sufficiently large, the number of such values is at most N^{1/k+ε}. The proofs use a factorization identity (t−t_0)P_k(t)=P_{k+1}(t)−P_{k+1}(t_0), a divisor-pair splitting, the external theorem of Bourgain–Demeter (Theorem 1.1), and Wigert's bound on d(q). The paper is written as a short comment and explicitly says both results should be known.

Significance. If the cited Bourgain–Demeter theorem is exactly as stated in Theorem 1.1, then the paper's results are correct, simple corollaries. Theorem 1.4 is a clean near-optimal uniform bound for polynomial values in arithmetic progressions with polynomially growing step, and Theorem 1.2 is a modest extension under a divisor-count condition. The paper's strength is that it is concise and identifies a black-box use of a deep external theorem plus an elementary divisor estimate. Its limitations are the reliance on an unverified attribution and a misleading induction framing. The contribution is not a new method but a short observation; this is acceptable for a comment if the external theorem is quoted correctly.

major comments (2)

[§1, Theorem 1.1 and §2, proofs of Theorems 1.2/1.4] The abstract says that Bourgain–Demeter [3] proved bounds for progressions 'whose step has O(1) many divisors,' but Theorem 1.1 attributes to [3, Thm 0.1] the stronger bound |{t: P_k(t)∈{a+q,...,a+Nq}}| ≲ d(q)^{k−1} N^{1/k} for all q. The proofs of Theorems 1.2 and 1.4 rely on this stronger form for arbitrary divisors q_2 of q. If [3, Thm 0.1] is only the O(1)-divisor case, the central upper bounds collapse because q_2 may have many divisors. Please quote the exact statement of [3, Thm 0.1] or prove the d(q)^{k−1} version.
[§2, proof of Theorem 1.2 (also Theorem 1.4)] The proof claims induction on k, but in the second case the estimate O(d(q_2)^{k−1} N^{1/(k+1)}) is attributed to 'the induction hypothesis'. The induction hypothesis of Theorem 1.2 for degree k would give N^{2/k}, not this bound. The bound used is exactly Theorem 1.1 applied to the degree-k polynomial P_k. Thus the proof is not an induction; it is a direct application of Theorem 1.1. The paper's stated claim that the 'same induction argument' as [3] can be used is not supported. Rewrite the proof as a direct corollary of Theorem 1.1, or actually carry out the Bourgain–Demeter induction step to justify the claim.

minor comments (5)

[§2, proof of Theorem 1.2] The displayed factorization is numbered (2), but the text refers to 'the first equation in (3)' when it should be (2).
[§2, proof of Theorem 1.4] Typo: 'we all the details' should be 'we add all the details'. Also, 'E-mail adress' should be 'E-mail address'.
[§2, proof of Theorem 1.2] The phrase 'We may assume the statement holds for k > 1' is awkward; it should be 'Fix k ≥ 1 and assume the statement holds for degree k; let P_{k+1} be a polynomial of degree k+1.'
[§1, Theorem 1.2] In the induction step for degree k+1, the proof uses the assumption d(q) ≲ N^{1/(k(k+1))}, while the theorem for degree k+1 states d(q) ≲ N^{1/((k+1)(k+2))}. The latter implies the former, so the argument is valid, but the monotonicity should be stated explicitly.
[References] Reference [3] is an arXiv preprint; if it has been published, update the citation. Also, the reference list has a minor typo in Hajdu and Papp's title: 'Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser: A-Mat.' should be 'Ser. A Mat.'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bound is an external-theorem corollary; the 'induction hypothesis' label is a naming error, not a logical loop.

full rationale

The paper derives upper bounds for the number of k-th powers in arithmetic progressions by combining two external inputs: Bourgain–Demeter [3, Thm 0.1] (quoted as Theorem 1.1) and Wigert's divisor estimate [8]. No parameter is fitted from the data being predicted, no self-citation carries the argument, and no uniqueness theorem from the authors' own work is imported. The proof of Theorems 1.2 and 1.4 calls the strong bound O(d(q2)^{k-1} N^{1/(k+1)}) 'the induction hypothesis,' but the quantitative form used is exactly the external Bourgain–Demeter bound, not the theorem being proved; this is a mislabelling of an external theorem, not a circular step. The final divisor estimates (Theorem 1.2's d(q) assumption, Theorem 1.4's use of Wigert) fold d(q)^k into N^ε independently. To the extent that the stronger d(q)-dependent version of the Bourgain–Demeter result is not explicitly restated or verified, that is a correctness or reproducibility concern about an external source, not circularity. The paper is self-contained relative to its stated external assumptions, and its contribution is an honest 'easy observation' corollary.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities are introduced. The load-bearing inputs are two external theorems (Bourgain–Demeter and Wigert) plus standard algebra; the paper's contribution is the observation that these combine to cover larger step sizes.

assumptions (3)

domain assumption Bourgain–Demeter Theorem 0.1 ([3]): for every degree-k integer polynomial P_k, |{t:P_k(t)∈{a+q,...,a+Nq}}| ≲ d(q)^{k-1} N^{1/k}.
Used as the engine of both proofs, though the paper calls it 'the induction hypothesis'. All new divisor estimates merely control the extra d(q) factors around this external bound.
standard math Wigert's divisor bound: max_{1≤n≤x} d(n) = x^{(log 2 + o(1))/log log x}.
Invoked in the proof of Theorem 1.4 to absorb d(q)^k into N^ε when q≤N^r.
standard math Polynomial factor theorem: P_{k+1}(t) − P_{k+1}(t0) = (t−t0)P_k(t) for some integer-coefficient polynomial P_k.
This factorization is the starting point of both induction proofs and leads to the splitting (t−t0)=n1 q1, P_k(t)=q2 n2 with q1q2=q.

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Cite this review

Pith. "Pith review of A comment on the number of $k$-th powers inside arithmetic progressions." pith.science (2026). https://pith.science/paper/NDM4JEQW

@misc{pith2026260715895,
  author       = {Pith},
  title        = {Pith review of: A comment on the number of $k$-th powers inside arithmetic progressions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NDM4JEQW}},
  note         = {Machine review of arXiv:2607.15895}
}

abstract

In \cite{BD} Bourgain and Demeter found sharp upper bounds for the number of $k$-th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors. We make the easy observation that the same arguments are still valid if the step does not grow too rapidly in relation to the length of the progression. Furthermore, we give sharp bounds for the number of $k$-th powers among the first $N$ terms for $N$ large enough. Both results should be known. Nevertheless, we add to the literature.

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