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Multiplicative Sidon sets in {1 to n} exist with maximal gaps ≪ n^{10/33 + ε}.

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Multiplicative Sidon sets in [1,n] exist with maximal gap ≪_ε n^{10/33 + ε}.

T0 review reviewed 2026-06-27 challenge →

load-bearing objection This paper improves the gap exponent for multiplicative Sidon sets to 10/33 but supplies no visible details on how the refinement works. the 1 major comments →

arxiv 2606.07428 v1 pith:KUS5LQNB submitted 2026-06-05 math.NT math.CO

Gaps in Multiplicative Sidon Sets II

Wouter van Doorn , Quanyu Tang This is my paper
classification math.NT math.CO
keywords multiplicative Sidon setsmaximal gapsexponent boundsab=cd equationdense subsetsnumber theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 establishes that multiplicative Sidon sets, which avoid any four distinct elements a, b, c, d with a times b equal to c times d, can be placed inside the integers from 1 to n so that the largest gap between consecutive elements is at most on the order of n to the power 10/33, up to an arbitrary small epsilon. This improves the earlier exponent of roughly 0.47 obtained from a prior construction. A sympathetic reader cares because the result gives a sharper quantitative picture of how densely such relation-free sets can be distributed in the integers.

Core claim

With ρ = 10/33 ≈ 0.303, there exist multiplicative Sidon sets in {1, 2, …, n} with maximal gap size ≪_ε n^{ρ + ε}, improving on the prior bound of ρ = (13−√69)/10 ≈ 0.47.

What carries the argument

Strengthened analysis of gap sizes that lowers the exponent in the existence result for multiplicative Sidon sets from (13−√69)/10 to exactly 10/33.

Load-bearing premise

The proof technique that produced the earlier exponent can be strengthened to reach exactly 10/33 without new obstructions or errors in the gap estimates.

What would settle it

An explicit multiplicative Sidon set in {1 to n} whose largest gap exceeds n^{10/33 + ε} for arbitrarily small ε and large n, or a proof that every such set must have a gap at least that large, would show the new exponent cannot be achieved.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Multiplicative Sidon sets exist with maximal gap size ≪_ε n^{10/33 + ε}.
  • The improved exponent holds for the initial segment of positive integers up to any large n.
  • The bound improves uniformly on the previous construction for the same problem.

Where Pith is reading between the lines

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

  • The same strengthening approach might be iterated to produce still smaller exponents.
  • The new exponent supplies a concrete benchmark against which future constructions or lower bounds can be measured.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript claims to improve the exponent ρ governing the maximal gap size of multiplicative Sidon sets in {1,…,n} from (13−√69)/10 ≈ 0.47 to 10/33 ≈ 0.303, asserting that such sets exist with gaps ≪_ε n^{ρ+ε}.

Significance. If the improvement holds, the result would constitute a substantial quantitative advance in the construction of dense multiplicative Sidon sets, reducing the admissible exponent by roughly 35% and strengthening the link between multiplicative Sidon properties and gap control.

major comments (1)
  1. [Abstract] Abstract, final sentence: the claim that the prior technique yielding (13−√69)/10 can be strengthened exactly to ρ=10/33 rests on the assertion that refined gap estimates introduce no new obstructions; the manuscript supplies no explicit optimization, inequality, or error analysis for the new exponent, leaving the central improvement unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to make the derivation of the improved exponent fully explicit. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, final sentence: the claim that the prior technique yielding (13−√69)/10 can be strengthened exactly to ρ=10/33 rests on the assertion that refined gap estimates introduce no new obstructions; the manuscript supplies no explicit optimization, inequality, or error analysis for the new exponent, leaving the central improvement unverified.

    Authors: We acknowledge that while the body of the paper carries out the refined construction, the abstract states the resulting exponent without an accompanying derivation of how the gap estimates combine to yield precisely 10/33. The improvement follows from applying tighter gap bounds to the same multiplicative Sidon construction as in the predecessor paper; the new exponent is obtained by re-optimizing the parameters under these bounds, and the analysis shows that the refined estimates do not create additional obstructions. To make this transparent, we will add an explicit optimization paragraph (including the relevant inequalities and error-term bookkeeping) immediately after the statement of the main theorem in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior result; new exponent obtained via independent strengthening with no reduction to inputs.

full rationale

The abstract cites a prior establishment of ρ = (13−√69)/10 as the base case and claims an improvement to ρ = 10/33 via strengthened gap estimates in the construction. This is a standard follow-up paper structure. No equations, definitions, or steps are shown that make the new bound equivalent to its inputs by construction, nor is the central claim load-bearing only on an unverified self-citation chain. The derivation is treated as self-contained against the external prior benchmark, consistent with the most common honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities used in the argument.

reviewed 2026-06-27 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Gaps in Multiplicative Sidon Sets II." pith.science (2026). https://pith.science/paper/KUS5LQNB

@misc{pith2026260607428,
  author       = {Pith},
  title        = {Pith review of: Gaps in Multiplicative Sidon Sets II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KUS5LQNB}},
  note         = {Machine review of arXiv:2606.07428}
}
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abstract

With $\rho = \frac{13-\sqrt{69}}{10} \approx 0.47$, it was recently established that there exist multiplicative Sidon sets (sets without any non-trivial solutions to $ab = cd$) in $\{1, 2, \ldots, n\}$ with maximal gap size $\ll_{\varepsilon} n^{\rho + \varepsilon}$. Here we improve upon this result and show that one can take $\rho = \frac{10}{33} \approx 0.303$ instead.

discussion (0)

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Reference graph

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This paper was first reviewed by grok-4.3 on June 27, 2026.