REVIEW 2 major objections 5 minor 9 references
Is it true that most sets are Sidon?
T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read Topologically, most subsets of N are not Sidon sets, and every B_h[g] family is meager.
desk verdict The answer to the title is right, but the printed proof of Theorem 1.1 has a genuine gap; the paper deserves refereeing after small repairs. 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 argument is carried by the representation-counting function $r_{A,h}(n)$, which counts unordered $h$-tuples from $A$ summing to $n$; a $B_h[g]$ set is exactly one with $r_{A,h}$ bounded by $g$. The genericity theorems are proved through the Banach–Mazur game on the Cantor space: Player II answers each open constraint by appending a long consecutive block of integers to $A$ and leaving a long gap before it, which simultaneously forces some $h$-fold sums to be absent and others to appear many times, yielding the oscillation in Theorem 1.2 and the perturbation resistance in Theorem 1.3. The finite-set result uses a different mechanism: the complement of the good $n$-tuples in $X^n$ is a finite union of intersections of hyperplanes, and each hyperplane is closed with empty interior because it is the kernel of a nonzero continuous linear functional.
What would settle it
Recompute the lower bound for $|A \cap (k_m, y_m]| / y_m$ from the game construction in Theorem 1.3: if there is a choice of $y_m$ (such as $y_m = (m+1) k_m$) that makes the bound $1 - 1/(m+1)$ valid for all $m$, the proof is repaired; if not, constructing $A$ from the strategy and checking whether the claimed inequality fails on infinitely many $m$ would settle that the theorem as stated is unproved.
Extended reading notes
Core claim
The central discovery is that $B_h[g]$ sets are topologically negligible in the Cantor-space topology on $\mathbb{P}(\mathbb{N})$. Theorem 1.1 proves each family $B_{h,g}$ is closed with empty interior, so the union over all $h$ and $g$ is meager. Theorem 1.2 strengthens this by showing that, for any divergent $f(n) = o(n^{h-1})$, a comeager set of $A \subseteq \mathbb{N}$ satisfies $\liminf_n r_{A,h}(n) = 0$ and $\limsup_n r_{A,h}(n)/f(n) = \infty$. Theorem 1.3 adds a robustness statement: the family of $A$ that admit some $B_h[g]$ set $B$ with lower density $d_\star(A \triangle B) < 1$ is meager. Finally, Proposition 1.4 inverts the picture: when $X$ is a real or complex topological vector space and the cardinality is fixed at $n$, the $n$-element $B_h[g]$ subsets form an open dense (hence comeager) subset of $X^n$, so among finite sets of a given size the property is generic rather than negligible.
Load-bearing premise
The perturbation theorem rests on the claim that after forcing the block $(k_m, y_m]$ into $A$, its density within $[0, y_m]$ is at least $1 - 1/(m+1)$, but the stated choice $y_m > m k_m$ only guarantees the weaker level $1 - 1/m$, so the proof needs an extra density argument or an index shift to go through.
Editorial extensions
If this is right
- The family of all $B_h[g]$ sets for some $h \geq 2$ and $g \geq 1$ is meager, so in particular the Sidon sets form a topologically negligible subset of $\mathbb{P}(\mathbb{N})$.
- For every $f(n) = o(n^{h-1})$, a comeager set of subsets $A$ has arbitrarily long gaps in its $h$-fold sums ($\liminf r_{A,h}(n) = 0$) and arbitrarily large spikes ($\limsup r_{A,h}(n)/f(n) = \infty$).
- A comeager set of $A \subseteq \mathbb{N}$ has no close $B_h[g]$ approximation in the sense of lower density of the symmetric difference being less than $1$.
- For any topological vector space $X$ and fixed cardinality $n$, the $n$-element $B_h[g]$ subsets are open dense in $X^n$, so among finite configurations the property is typical.
- The quantitative estimate for the full interval, $r_{\{0,1,\dots,n\},h}(n) \sim n^{h-1}/(h!(h-1)!)$, marks the threshold: anything asymptotically larger than $n^{h-1}$ can be beaten by the generic set's spikes.
Reading between the lines
- The paper works entirely in the category of Baire genericity; a natural testable extension is to check whether the same conclusions hold for almost every subset under the fair-coin product measure on $\{0,1\}^{\mathbb{N}}$, where the independence of bits would likely produce similar oscillation phenomena.
- The reversal in Proposition 1.4 suggests that Sidon-type constraints are purely asymptotic: every finite configuration can be realized inside a set with the property, so the topological rarity in $\mathbb{P}(\mathbb{N})$ is driven entirely by infinite-tail behavior.
- The sharpness condition $f(n) = o(n^{h-1})$ implies a scaling law for generic deviation: the number of representations of $n$ as a sum of $h$ elements of a generic set fluctuates between zero and arbitrarily large multiples of any prescribed sub-$n^{h-1}$ rate.
- Read as a statement about convolution, Theorem 1.2 says the $h$-fold additive convolution of a generic $0$-$1$ sequence is unbounded in a sparse way; this could be connected to combinatorial models where sparsity forces unusual limit points.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the family of Bh[g] subsets of the nonnegative integers (with Sidon sets as the case h=2, g=1) from the topological point of view of the Cantor space P(N)={0,1}^N. Theorem 1.1 claims that each Bh,g is closed and has empty interior, hence is meager. Theorem 1.2 states that for every f(n)=o(n^{h-1}), the generic A satisfies liminf r_{A,h}(n)=0 and limsup r_{A,h}(n)/f(n)=∞. Theorem 1.3 claims that the family of A that are within lower-density distance <1 from some Bh[g] set is meager. Proposition 1.4 claims an open dense (hence comeager) statement for n-element Bh[g] subsets of a topological vector space. The proofs use basic cylinders, the Banach–Mazur game, and the Erdős–Lehner asymptotic (1).
Significance. If the gaps noted below are repaired, the paper gives a clean and convincing topological answer: almost all subsets of N are not Sidon or Bh[g], and the generic behavior of representation functions is extreme. Theorem 1.2 is optimal in view of the asymptotic (1), and Theorem 1.3 shows a form of stability under small perturbations of lower density. The arguments are short, transparent, and rely only on standard external results; there are no free parameters and no circular dependencies. The paper is therefore a useful contribution to the combinatorial number theory literature. As printed, however, two load-bearing steps in the proofs of Theorems 1.1 and 1.3 are not valid, so the central claims are not yet fully established.
major comments (2)
- [§2, proof of Theorem 1.1] The empty-interior argument is not valid as written. The contradiction only rules out the existence of a finite F0 such that every superset A⊇F0 belongs to Bh,g, i.e. an upward-closed open set of the form {A:F0⊆A}. A nonempty basic cylinder [F]_k={A:A∩[0,k]=F} need not contain any set of that form; for example, C={A:A∩[0,1]={0}} contains no set {A:F0⊆A} because any such set contains an A with 1∈A. Thus the proof does not establish that every cylinder contains a point outside Bh,g. The gap is repairable: given [F]_k, set x0=k+1 instead of 1+max F0; the same construction yields A∈[F]_k with r_{A,h}(h(x0+g))≥g+1. This repair must be inserted for the main meagerness claim to follow.
- [§2, proof of Theorem 1.3] The inequality 'ym > m km implies |A∩[1,ym]|/ym ≥ 1 - 1/(m+1)' is false. Since A∩[0,ym]=Fm∪{km+1,...,ym}, the ratio equals (ym-km)/ym = 1 - km/ym, and the condition ym > m km gives only km/ym < 1/m, hence 1 - km/ym > 1 - 1/m, which is strictly smaller than 1 - 1/(m+1). Consequently the lower bound '≥ c' with c = 1/m0 - 1/(m0+1) does not follow for all m≥m0. The proof is repairable by choosing ym > (m+1)km, or by applying the estimate for all sufficiently large m after a finite initial segment is discarded, but as printed this is a load-bearing gap.
minor comments (5)
- [Title, Abstract] The abstract 'No.' is a stylistic oddity that should be replaced by a genuine one-sentence summary for a journal publication.
- [Throughout] There are several typos: 'pertubations' in Section 1, 'we define define' in the proof of Theorem 1.2, 'this is obvious is' in the proof of Theorem 1.1, and 'a winning for Player II' in Theorem 1.2.
- [§2, proof of Theorem 1.3] After showing that each M_{h,g} is comeager for fixed h and g, the proof should explicitly say that taking the countable intersection over h≥2 and g≥1 yields the theorem; the countable intersection of comeager sets is comeager.
- [§2, Proposition 1.4] The conclusion that the family of n-element Bh[g] subsets is a 'dense Gδ subset of X^n' for arbitrary topological vector spaces X requires X^n to be a Baire space. The weaker conclusion 'comeager' follows without any Baire assumption, and the proof establishes only that. Either add a Baire assumption on X or weaken the final sentence accordingly.
- [§2, Proposition 1.4] The identification of n-element subsets with vectors in X^n is not one-to-one because of permutations; the statement is harmless since the set V is permutation-invariant, but it should be phrased carefully.
Circularity Check
No significant circularity: the meagerness and comeagerness arguments rest on external textbook theorems (Kechris, Erdős–Lehner, Bogachev–Smolyanov); the two self-citations are motivational, not load-bearing.
full rationale
Walking the derivation chain shows no step that reduces to its own inputs. Theorem 1.1: Bh,g is defined by the standard bound r_{A,h}(x) ≤ g; closedness is proved directly from that definition, and the empty-interior argument constructs, from a finite F0, a superset A with at least g+1 representations of h(x0+g), so the meagerness conclusion is not assumed in the premise. Theorem 1.2: the strategy for Player II uses Kechris's Banach–Mazur theorem [4, Theorem 8.33] and the external Erdős–Lehner asymptotic (1) only to guarantee existence of the integers tm; neither source encodes the claimed liminf = 0 / limsup = ∞ behavior. Theorem 1.3 runs the same Banach–Mazur strategy with a counting argument; Proposition 1.4 is proved from finite intersections of hyperplanes with empty interior, citing only Bogachev–Smolyanov hyperplane facts. The self-citations are not load-bearing: [5] (Leonetti 2023) is cited only as the 'analogue direction' for the perturbation claim, and [6] (Leonetti–Tringali 2020) only for the definition of lower density; the extension of Nathanson's preprint [7] is proved in the paper, not imported from it. The reviewer-flagged gaps — Theorem 1.1's contradiction handles only cylinders of the form {A : F0 ⊆ A} rather than an arbitrary basic cylinder {A : A ∩ [0,k] = F}, and Theorem 1.3 derives |A∩(k_m,y_m]|/y_m ≥ 1 − 1/(m+1) from y_m > m k_m, which only yields 1 − 1/m — are correctness risks that may invalidate the printed proofs; they are not circularity, since neither gap consists in assuming the target result, fitting a parameter renamed as a prediction, or importing the conclusion from a self-citation. The post-Proposition 1.4 dense Gδ remark silently assumes X^n is Baire, again a correctness gap rather than a circular step. The paper is self-contained against external benchmarks, so the honest finding is no significant circularity.
Assumptions & free parameters
assumptions (5)
- standard math The Cantor space {0,1}^N is a Polish space; meager sets form a sigma-ideal; the Banach-Mazur game characterizes comeager sets (Kechris Theorem 8.33).
- standard math Asymptotic formula r_{0,1,...,n,h}(n) ~ n^{h-1}/(h!(h-1)!) as n tends to infinity, equation (1), cited from [3], [9], and [8].
- standard math For a nonzero continuous linear functional on a Hausdorff topological vector space, its kernel is closed and has empty interior.
- domain assumption X^n is a Baire space, for example when X is a Banach or Frechet space, so a countable intersection of dense open sets is dense G_delta.
- standard math Definition and basic properties of lower asymptotic density d_*, including that d_*(A triangle B) < 1 implies the symmetric difference is eventually smaller than n(1 - 1/m0).
Cite this review
Pith. "Pith review of Is it true that most sets are Sidon?." pith.science (2026). https://pith.science/paper/4SFEOUWD
@misc{pith2026250703413,
author = {Pith},
title = {Pith review of: Is it true that most sets are Sidon?},
year = {2026},
howpublished = {\url{https://pith.science/paper/4SFEOUWD}},
note = {Machine review of arXiv:2507.03413}
}
read the original abstract
No.
Reference graph
Works this paper leans on
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[5]
P. Leonetti, Almost all sets of nonnegative integers and their small perturbations are not sumsets, Proc. Amer. Math. Soc.151 (2023), no. 9, 3681–3689
work page 2023
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[1]
V. I. Bogachev and O. G. Smolyanov, Topological vector spaces and their applications, Springer Monographs in Mathematics, Springer, Cham, 2017
work page 2017
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[2]
J. Cilleruelo, I. Ruzsa, and C. Vinuesa,Generalized Sidon sets, Adv. Math.225 (2010), no. 5, 2786–2807
work page 2010
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[3]
P. Erdös and J. Lehner,The distribution of the number of summands in the partitions of a positive integer, Duke Math. J.8 (1941), 335–345
work page 1941
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[4]
A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995
work page 1995
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[6]
P. Leonetti and S. Tringali,On the notions of upper and lower density, Proc. Edinb. Math. Soc. (2) 63 (2020), no. 1, 139–167
work page 2020
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[7]
M. B. Nathanson, Bh-sets of reals and complex numbers, preprint, last updated: Mar 05, 2025 (arXiv:2502.21272)
work page Pith review arXiv 2025
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[8]
, Partitions with parts in a finite set, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1269–1273
work page 2000
Show all 9 references
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[9]
J. L. Ramírez Alfonsín, The Diophantine Frobenius problem, Oxford Lecture Series in Mathematics and its Applications, vol. 30, Oxford University Press, Oxford, 2005. Is it true that most sets are Sidon? 7 Department of Economics, Università degli Studi dell’Insubria, via Monte...
2005
Reviewed August 6, 2026 · model on record in the stance chip above.
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