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On the Thickness of Infinite Generalized Sidon Sets, II

T0 review · 2 major / 6 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Every infinite even-order generalized Sidon set is thinner than a sharp multiple of (n/log n) to the power 1/h.

desk verdict Explicit constant for even-order infinite B_h thickness, cleanly upgrading Chen, but the load-bearing energy-to-liminf step lives in an unrefeered Part I. read the letter →

arxiv 2607.23795 v1 pith:M5KPGWC6 submitted 2026-07-26 math.CO math.NT

Kevin O'Bryant This is my paper · ORCID
classification math.COmath.NT MSC 05B1011B8311B05
keywords B_h-setsSidonsetsthicknesssumsetsliminfboundsadditivebasesevenorder
topics The Collatz 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

A B_h-set is a set of nonnegative integers in which every sum of h elements, written in nondecreasing order, is unique. For ordinary Sidon sets (h=2) it has long been known that the counting function cannot stay as large as a constant times sqrt(n/log n) forever. This paper supplies an explicit constant that works for every even h: the liminf of A(n) divided by (n/log n)^{1/h} is at most a concrete expression built from pi, log 2 and gamma values. The bound improves earlier qualitative finiteness results and recovers the author's earlier Sidon-set constant when h=2. A sympathetic reader cares because the constant is sharp enough to be compared with known constructions and because the argument isolates exactly where the even-order hypothesis is used.

What carries the argument

Control of the squared block counts of the k-fold sumset S=kA (k=h/2) via a multiset difference stratification and Jia-type counting of Phi tuples, then conversion of that energy bound into a liminf upper bound by a weighted block-energy lemma.

What would settle it

Exhibit an infinite B_h-set (h even) whose counting function satisfies A(n) > C (n/log n)^{1/h} for every n larger than some n0, where C is strictly larger than the constant stated in Theorem 1.

Watch

Extended reading notes

Core claim

For every even positive integer h and every B_h-set A, the liminf as n tends to infinity of A(n) divided by the h-th root of n/log n is at most (pi/log 2 times Gamma(1+h/2)^2 over Gamma(1+1/h)^h) raised to the power 1/h.

Load-bearing premise

The conversion from a uniform upper bound on the sum of binomial block counts into an explicit liminf constant is taken as already proved in an earlier paper and is not re-established here.

Editorial extensions

If this is right

  • The corresponding limsup lower bound on the n-th term a_n is at least (log 2/pi) times h Gamma(1+1/h)^h over Gamma(1+h/2)^2, times n^h log n.
  • The same constant specializes, when h=2, to the Sidon-set thickness bound previously obtained by the author.
  • Any future improvement of the block-energy constant c=1/2 immediately improves the explicit thickness constant for every even h.
  • The method does not yet yield an analogous explicit liminf for odd h, leaving Jia's conjecture open.

Reading between the lines

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

  • Because every B_h-set is automatically a B_{h-1}-set, the even-order bound already gives a (weaker) thickness statement for odd order by reduction, though the paper treats that as artificial.
  • The asymptotic of the constant is h/(2e); numerical plots in the paper show a surprising minimum at h=4, suggesting that thickness is hardest to force precisely at order 4.
  • If a matching construction ever reaches the same constant, the liminf would be settled exactly for even h.
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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

2 major / 6 minor

Summary. The manuscript proves that for every even positive integer h and every B_h-set A of nonnegative integers, liminf A(n)/(n/log n)^{1/h} is bounded by the explicit constant (π/log 2 · Γ(1+h/2)²/Γ(1+1/h)^h)^{1/h}. This is the first explicit constant for this liminf for general even h; Chen (1993) had shown finiteness without a constant, and the h=2 case recovers the √(4/log 2) bound from the author's Part I. The proof is by contradiction: assuming A(x) ≥ τ(x/log x)^{1/h} with τ above the claimed constant, a new Lemma 8 lower-bounds liminf S(m)/√(m/log m) for the k-fold sumset S = kA (k = h/2) via a lattice-point count and a Dirichlet integral; Lemma 11 produces a shift t* whose block counts satisfy Σ C(F_ℓ,2) ≤ N/2 + o(N) by averaging over shifts and stratifying difference pairs by |V_s ⋒ V_s'|, using a quantitative form of Jia's Lemma 9; and Lemma 6 (imported from Part I) converts the block-energy bound into liminf S(m)/√(m/log m) ≤ √(4/log 2). Comparing the two bounds on the same liminf yields the contradiction.

Significance. If correct, this is a solid contribution: the first explicit, parameter-free constant for the thickness of infinite B_h-sets for all even h, improving on 35-year-old finiteness results, and it eliminates Jia's growth-regularity hypothesis A(N²) = O(A(N)²) by separating the block count M from the block width N (M = N/log³ N) — a genuine technical advance over Erdős–Helm–Jia–Chen. The constant is fully explicit and falsifiable, the h=2 specialization correctly recovers Part I's √(4/log 2), and the asymptote h/(2e) with minimum at h=4 is a clean, checkable statement. The argument is short and, apart from one imported lemma, self-contained and verifiable line by line; I checked the Dirichlet integral evaluation, the Γ(1+1/h) = (1/h)Γ(1/h) reconciliation, the final algebra, and Corollary 2 (including the factor h from log n ≈ (1/h) log x), all of which are correct. The one soft point is that the constant's log 2 factor enters entirely through Lemma 6, which is stated but not proved here.

major comments (2)
  1. [§3, Lemma 6; invoked in §4] Lemma 6 is stated with hypotheses (i)–(iii) but no proof, and no precise citation (not even a theorem number) into Part I [12], which is listed only as 'Preprint.' This is load-bearing: the entire explicit content of Theorem 1 and Corollary 2 inherits the factor √(8c/log 2), and the log 2 — the only transcendental input beyond the Dirichlet integral — is produced nowhere in this manuscript. A naive Cauchy/extremal-profile computation with F_ℓ ∝ 1/√ℓ yields only Σ F_ℓ² ≲ λ²N/4, i.e. √(8c) with no log 2, so the constant depends on a specific argument in Part I that the reader cannot audit. Moreover, Part I used M = N/log N while this paper needs M = ⌊N/log³ N⌋; the manuscript asserts that hypotheses (i)–(iii) suffice but gives no indication that Part I's proof runs under these general hypotheses rather than for its specific M. Please either include a proof of Lemma 6 (an appendix is fine)
  2. [§3, proof of Lemma 11, Eqs. (7)–(10)] The left-hand side of the displayed chain changes from Σ_{ℓ=1}^{M} C(F_ℓ,2) in (7) to Σ_{ℓ=1}^{M−1} C(F_ℓ,2) in (8), and the proof concludes with the M−1 sum, while the lemma's statement — and the application to Lemma 6 in §4, which needs exactly the blocks ℓ = 1,…,M — concerns the M sum. Since C(F_M,2) ≥ 0, an upper bound on the M−1 sum does not imply one on the M sum, so as written the proof does not establish the stated lemma. The repair is immediate (rerun the argument with M+1 in place of M, or restate the lemma with M−1 and note Lemma 6's hypotheses are asymptotic and insensitive to this shift), and no constant is affected, but the mismatch must be fixed.
minor comments (6)
  1. [§3, proof of Lemma 8, final display] In the three-line chain ending the proof, the middle relation is printed as '≤' but must be '≥': (1 − 1/log u_0)^k ≥ 1 − k/log u_0 ≥ 1 − k/log log m since u_0 ≥ log m. As printed the chain reads N*(m) ≥ · ≤ ·, which is confusing.
  2. [§3, Lemma 11 statement] The hypothesis allows τ ≥ 0, but the proof's ratio A(W)^r/A(N)^{2r} → 0 requires τ > 0. State τ > 0. Also 'let k := h/2' begins with a lowercase letter after a period.
  3. [§3, Lemma 6, hypothesis (iii)] 'B(N) = o(√N log N)' is ambiguous between o(√(N log N)) and o(√N · log N); in §4 it is verified as O(N^{1/2}) = o(√(N log N)). Please disambiguate with parentheses.
  4. [§1.2] Typos: 'with substantials detours' and 'artifical manner.'
  5. [References] [12] (Part I) is cited only as 'Preprint'; please add an arXiv identifier and, given that Lemma 6 and the weighted Cauchy inequality are imported from it, precise internal pointers (section/theorem numbers).
  6. [§1, Corollary 2] The display lacks a terminal period. It would also help the reader to note in one line where the factor h comes from (log a_n ≈ (1/h) log(a_n/(log a_n)^{...}) inversion), since it is not the naive reciprocal of Theorem 1's constant.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is a genuine proof-by-contradiction whose constant is derived from Dirichlet integrals, B_h uniqueness, and a block-energy bound, not from fitting or assuming the claim.

full rationale

The derivation chain is: assume A grows at least as fast as τ(n/log n)^{1/h} with τ larger than the target constant; form S=kA (k=h/2); obtain a lower bound on liminf S(m)/√(m/log m) via lattice-point counting and a Dirichlet integral (Lemma 8); obtain an upper bound on the same liminf by controlling pairwise differences of S in sliding blocks of width N (Lemma 11 gives ∑ binom(F_ℓ,2) ≤ N/2+o(N)), then invoking the general energy-to-liminf conversion (Lemma 6 with c=1/2). Comparing the two bounds on liminf S contradicts the choice of τ and yields the explicit Gamma constant. Nothing is defined in terms of the target liminf; no parameter is fitted to data; no uniqueness theorem is imported to forbid alternatives; and the final algebra is ordinary comparison of independently derived bounds. Dependence on the author's Part I for the statement of Lemma 6 (and weighted Cauchy) is ordinary sequential publication of a lemma, not a circular reduction of Theorem 1 to itself. The paper is self-contained as a combinatorial argument once that lemma is granted.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper is pure analytic number theory / additive combinatorics. It rests on the definition of B_h-sets, standard Gamma and Dirichlet-integral identities, stars-and-bars multiset counts, and one substantial black-box lemma (Lemma 6) imported from Part I. No empirical free parameters. Technical sets T and Φ are bookkeeping devices, not new physical entities.

assumptions (5)
  • domain assumption Definition of a B_h-set: all nondecreasing h-fold sums from A are distinct.
    Load-bearing definition used in every uniqueness argument (Lemmas 3–5, 9–11).
  • standard math Dirichlet integral evaluation ∫_{u_i>0, sum u_i≤m} ∏ u_i^{1/h−1} du = 2 Γ(1/h)^k / √π · √m (Whittaker–Watson).
    Invoked in the proof of Lemma 8 to convert the lattice-point count into a closed form.
  • ad hoc to paper Lemma 6: if block energies ∑ binom(F_ℓ,2) ≤ c N + o(N) under the stated M(N) hypotheses, then liminf B(m)/√(m/log m) ≤ √(8c/log 2).
    Stated without proof in this manuscript; treated as known from Part I and fed c=1/2 to obtain the final constant.
  • standard math Stars-and-bars / multichoose bounds |X|^i/i! ≤ multichoose(|X|,i) ≤ (|X|+i)^i/i!.
    Used repeatedly to pass between sumset cardinalities and multiset counts (e.g., (2), (3), Lemma 11).
  • standard math Lemma 3 (multiset cancellation): if P⊎Q'=P'⊎Q with P∩Q=P'∩Q'=∅ then P=P' and Q=Q'.
    Elementary uniqueness lemma used to show P_0(d)≤1 and to run the induction in Jia's Φ-bound.
invented entities (1)
  • Φ(r,p,q;x,L) and T(p,q;x,L) counting sets
    purpose: Bookkeeping devices that let the author bound ordered tuples of A-elements together with disjoint multiset pairs of given sum-difference, following Jia.
    Purely combinatorial counters; no ontological claim beyond the paper's counting arguments.

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Pith. "Pith review of On the Thickness of Infinite Generalized Sidon Sets, II." pith.science (2026). https://pith.science/paper/M5KPGWC6

@misc{pith2026260723795,
  author       = {Pith},
  title        = {Pith review of: On the Thickness of Infinite Generalized Sidon Sets, II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5KPGWC6}},
  note         = {Machine review of arXiv:2607.23795}
}
abstract

A set $\mathcal{A}$ of nonnegative integers is a $B_h$-set if the sums $a_1+\cdots+a_h$ with $a_1\le\cdots\le a_h$ and $a_i\in\mathcal{A}$ are distinct; a $B_2$-set is a Sidon set. We prove that for every even $h$ and every $B_h$-set $\mathcal{A}$, \[ \liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt[h]{n/\log n}} \le \left(\frac{\pi}{\log 2} \cdot \frac{\Gamma(1+h/2)^2}{\Gamma(1+1/h)^{h}}\right)^{1/h}. \]

Figures

Figures reproduced from arXiv: 2607.23795 by the authors.

Figure 1
Figure 1. The constant of Theorem 1, together with its asymptote [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Schematic showing the relationships A(x) ≥ f(x) for x > n0, f(xj ) = j, and aj < xj . Each integer tuple (j1, . . . , jk) corresponds to the unit box Q i (ji − 1, ji ], and on that box ⌈yi⌉ = ji . Therefore N ∗ (m) = vol ( (y1, . . . , yk) ∈ (J0 − 1, ∞) k : X k i=1 x⌈yi⌉ ≤ m ) . The function f has an inverse on [e,∞) that is also continuous and strictly increasing. For ⌈yi⌉ ≥ J0, we have f −1 (yi + 1) ≥ f −1 (⌈yi⌉) … view at source ↗

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