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arxiv: 2604.24891 · v1 · submitted 2026-04-27 · 🧮 math.CO

Gap sets of random generalized numerical semigroups

Veronica Bitonti , Noah Kravitz This is my paper

Pith reviewed 2026-05-08 02:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords random numerical semigroupsgap setshyperboloid approximationgeneralized numerical semigroupssubset sumsprobabilistic combinatoricsadditive bases
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The pith

The gap set of a p-random generalized numerical semigroup in fixed dimension d is approximated by a shifted hyperboloid as p approaches zero.

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

This paper studies the structure of gaps in semigroups generated by random subsets of the d-dimensional non-negative lattice, where each point is included with small probability p. It establishes that these gaps, with high probability, lie within a region defined by the product of shifted coordinates being much less than p inverse times a power of the log. This provides a geometric description of how the additive closure fills space randomly, extending one-dimensional results and holding also for subset sums alone. Understanding this helps predict the typical size and shape of unfilled regions in random additive structures.

Core claim

The authors show that with high probability as p tends to zero, the gap set is well approximated by the set of points where the product (x1 + log p^{-1}) through (xd + log p^{-1}) is much smaller than p^{-1} times (log p^{-1})^{d+1}. The same holds when replacing the semigroup with its subset sums.

What carries the argument

The shifted hyperboloid region given by the product inequality on the coordinates shifted by log of one over p.

Load-bearing premise

The probabilistic approximation to the hyperboloid region becomes accurate as the sampling probability p tends to zero while keeping the dimension fixed.

What would settle it

Numerical experiments sampling A with very small p in d=2 or 3 and checking whether the boundary of the actual gap set deviates substantially from the predicted product region.

Figures

Figures reproduced from arXiv: 2604.24891 by Noah Kravitz, Veronica Bitonti.

Figure 1
Figure 1. Figure 1: The region R2(p, Z) lies under the curve (x1+log p −1 )(x2+log p −1 ) = Z. gap element that can be used to define the Frobenius number of a generalized numerical semi￾group, but the genus (number of gaps) is still a useful concept. The Erd˝os–R´enyi model in the multidimensional setting is defined just as in the 1-dimensional setting. Our main result provides a with-high-probability description of the gap … view at source ↗
Figure 2
Figure 2. Figure 2: The regions involved in the proof of Theorem 3.6 for m = 2. The “seed” set A0 lies in the small blue rectangle. Its set of subset sums FS(A0) contains many elements of the larger red rectangle. Next, FS(A) contains all of the points of the orange rectangle W′ (Y1, Y2). Finally, FS(A) contains all of the points of the (unbounded) yellow region W(Y1, Y2) \ W′ (Y1, Y2). Proof of Proposition 3.4. Let C ′ be 2m… view at source ↗
read the original abstract

For a fixed positive integer $d$ and a small real $p>0$, sample a $p$-random subset $A \subseteq \mathbb{Z}_{\geq 0}^d$, and let $S:=\langle A \rangle$ be the generalized numerical semigroup generated by $A$. We show that with high probability (as $p \to 0$), the gap set $\mathbb{Z}_{\geq 0}^d \setminus S$ is well approximated by the shifted hyperboloid region $$\{(x_1, \ldots, x_d) \in \mathbb{R}_{\geq 0}^d: (x_1+\log p^{-1}) \cdots (x_d+\log p^{-1})\ll p^{-1}(\log p^{-1})^{d+1}\}.$$ This generalizes work of the second author, Morales, and Schildkraut on the $1$-dimensional setting. We also obtain the same result with $S$ replaced by the set of subset sums of $A$.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript proves that for fixed d ≥ 1 and p → 0, if A is a random subset of ℤ≥0^d with each point included independently with probability p, then the gap set ℤ≥0^d ∖ ⟨A⟩ is asymptotically approximated (in a sense made precise in the paper) by the shifted hyperboloid region {(x1,…,xd) ∈ ℝ≥0^d : ∏(xi + log(1/p)) ≪ p^{-1} (log(1/p))^{d+1}} with high probability. An identical statement holds when ⟨A⟩ is replaced by the set of all subset sums of A. The argument generalizes the one-dimensional analysis of the second author et al. via direct probabilistic estimates on the reachability of lattice points.

Significance. If the stated high-probability approximation holds, the work supplies a concrete geometric description of the gaps in sparse random semigroups in any fixed dimension. The proof strategy—independent Bernoulli sampling together with scaling by log(1/p)—is standard for such models and yields a parameter-free limiting shape, which is a strength. The result is of interest to combinatorial number theory and probabilistic combinatorics on monoids.

major comments (2)
  1. [§4, Theorem 1.1] §4, Theorem 1.1 and the definition of 'well approximated': the precise metric (e.g., whether the symmetric difference has o(vol) measure, or whether the Hausdorff distance between the discrete gap set and the continuous region tends to zero after suitable scaling) is not stated explicitly in the theorem; without it the high-probability claim cannot be verified or falsified.
  2. [§5.2, Lemma 5.3] §5.2, Lemma 5.3 (the key reachability estimate): the error term in the probability that a lattice point outside the hyperboloid is reachable appears to be O(p^ε) for some ε>0, but the dependence on d is not tracked uniformly; when d is fixed this is harmless, yet the argument should record the d-dependence explicitly to confirm it remains o(1) as p→0.
minor comments (3)
  1. [Abstract and §2] Notation: the symbol ≪ is used without a quantitative definition; replace it by an explicit inequality involving a function ω(p)→∞ or state the implied constants.
  2. [§1] The 1-dimensional case is cited but the precise statement from Morales–Schildkraut is not recalled; a one-sentence reminder would help readers.
  3. [§6] Figure 1 (if present) or the illustrative plots in §6 should include the scaled hyperboloid boundary for visual comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and positive recommendation for minor revision. Below we address each major comment in turn, indicating the changes made to the manuscript.

read point-by-point responses

  1. Referee: [§4, Theorem 1.1] §4, Theorem 1.1 and the definition of 'well approximated': the precise metric (e.g., whether the symmetric difference has o(vol) measure, or whether the Hausdorff distance between the discrete gap set and the continuous region tends to zero after suitable scaling) is not stated explicitly in the theorem; without it the high-probability claim cannot be verified or falsified.

    Authors: We thank the referee for this observation. While the definition of 'well approximated' appears in the text preceding Theorem 1.1, we concur that the theorem statement would benefit from an explicit reference to the precise approximation metric. In the revised manuscript, we have modified the statement of Theorem 1.1 to include a parenthetical clarification of the sense in which the approximation holds (specifically, that the scaled symmetric difference has vanishing relative volume with high probability). This addresses the verifiability concern without altering the underlying result. revision: yes

  2. Referee: [§5.2, Lemma 5.3] §5.2, Lemma 5.3 (the key reachability estimate): the error term in the probability that a lattice point outside the hyperboloid is reachable appears to be O(p^ε) for some ε>0, but the dependence on d is not tracked uniformly; when d is fixed this is harmless, yet the argument should record the d-dependence explicitly to confirm it remains o(1) as p→0.

    Authors: We appreciate the referee's suggestion to make the d-dependence explicit. Although d is held fixed in the paper, recording the dependence clarifies that the error remains o(1) uniformly for fixed d. In the revised Lemma 5.3, we have added a remark tracking the constants' dependence on d, confirming that for any fixed d the probability bound is indeed o(1) as p → 0. No change to the main argument is required. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained probabilistic analysis

full rationale

The paper establishes the high-probability approximation of the gap set to the shifted hyperboloid region via direct probabilistic estimates on lattice-point reachability under independent Bernoulli sampling of A, with scaling by log(1/p). This extends the cited 1D case but does not reduce the d-dimensional claim to any fitted parameter, self-definition, or load-bearing self-citation chain. The prior work supplies only the base case; the new argument relies on standard sparse-random-model techniques that are independently verifiable and not equivalent to the target region by construction. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard facts about numerical semigroups and probabilistic limits; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math Standard properties of generalized numerical semigroups in d dimensions
    Used to define S and the gap set.
  • domain assumption High-probability convergence of gaps to the hyperboloid as p to 0
    Central to the main statement.

pith-pipeline@v0.9.0 · 8989 in / 1027 out tokens · 57306 ms · 2026-05-08T02:14:57.704356+00:00 · methodology

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Reference graph

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13 extracted references · 13 canonical work pages

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