Gaps of Binary Numerical Semigroups and of Binary Inclusion-Exclusion Polynomials
Pith reviewed 2026-05-20 14:13 UTC · model grok-4.3
The pith
Dominant pairs in linear modular permutations give complete gap descriptions for binary inclusion-exclusion polynomials and two-generated numerical semigroups.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By analyzing the dominant pair property of the linear permutation u n mod p, the paper provides a complete description of the gapsets of the binary inclusion-exclusion polynomials Q_{p,q} including the binary cyclotomic polynomials as a special case, and a complete description of all possible distances between consecutive elements of the numerical semigroup generated by p and q.
What carries the argument
The dominant pair property for pairs of integers (a,b) with respect to the linear permutation u·n mod p, defined by the max or min of the residues at a and b being strictly larger or smaller than those in between.
If this is right
- The gap sets of Q_{p,q} are explicitly determined by the dominant pairs in the associated permutation.
- All possible distances between consecutive elements in the numerical semigroup generated by p and q are classified completely.
- The results specialize to binary cyclotomic polynomials as a principal case.
- These descriptions allow direct computation of gaps without enumerating the full structures.
Where Pith is reading between the lines
- This modular permutation analysis might extend to inclusion-exclusion polynomials involving more factors.
- Similar techniques could address gaps in numerical semigroups generated by more than two elements.
- The dominant pair idea may connect to problems in uniform distribution or residue class behaviors.
Load-bearing premise
That the dominant pair property of the linear permutation is complete and directly sufficient to derive the full descriptions of the gapsets and distances.
What would settle it
Observing a specific pair of coprime integers p and q for which the gaps in the associated polynomial or semigroup distances do not match those predicted by the dominant pair analysis.
read the original abstract
Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\cdot n\pmod p$ of the residue system modulo $p$. Writing $\langle x\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that a pair of integers $(a,b)$ is a dominant pair of this permutation if either the inequality $\max(\langle ua\rangle_p,\langle ub\rangle_p)<\min_{a<n<b}\langle un\rangle_p$, or the inequality $\min(\langle ua\rangle_p,\langle ub\rangle_p)>\max_{a<n<b}\langle un\rangle_p$ hold. The main technical part of this work gives analysis of this property of linear permutations of residue systems. We then apply this analysis to the problems that motivated it, and give (i) complete description of the gapsets of binary inclusion-exclusion polynomials $Q_{\{p,q\}}$ (which include binary cyclotomic polynomials $\Phi_{pq}$ as its principal special case), and (ii) complete description of all possible distances between consecutive elements of a numerical semigroup $\langle p,q\rangle$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines a dominant pair (a,b) for the linear permutation n ↦ ⟨u n⟩_p (u coprime to p) via the two extremal inequalities on the images of a and b versus the images of the integers strictly between them. It supplies a combinatorial analysis of when such pairs occur and then applies the resulting classification to obtain explicit, exhaustive lists of the gapsets of the binary inclusion-exclusion polynomials Q_{{p,q}} (with binary cyclotomic polynomials Φ_{pq} as the main special case) and of all possible successive differences in the numerical semigroup ⟨p,q⟩.
Significance. If the central claims are correct, the work supplies parameter-free, explicit descriptions of two families of gaps that have previously been studied only through generating functions or case-by-case computation. The self-contained combinatorial treatment of dominant pairs, which translates directly into the two applications without fitted parameters or external benchmarks, is a clear strength and may serve as a template for similar problems in additive combinatorics.
minor comments (3)
- The notation ⟨x⟩_p is introduced on the first page but never explicitly recalled when it reappears in the statements of the main theorems; a one-sentence reminder would improve readability.
- In the application sections the authors list the possible gaps and distances but do not include a small worked example (e.g., p=5, q=8) that would let the reader verify the translation from dominant-pair data to the final lists.
- A few sentences in the introduction refer to “the problems that motivated it” without citing the specific earlier papers on gaps of ⟨p,q⟩ or on binary cyclotomic polynomials; adding those references would clarify the context.
Simulated Author's Rebuttal
We thank the referee for their positive and accurate summary of the manuscript, as well as the recommendation for minor revision. We are pleased that the combinatorial analysis of dominant pairs and its direct applications to gapsets and numerical semigroups are viewed as strengths. Below we address the major comments.
Circularity Check
No significant circularity; self-contained combinatorial derivation
full rationale
The paper introduces the dominant-pair property directly from the definition of the linear permutation u·n mod p and the residue function ⟨x⟩_p. It then performs an explicit combinatorial analysis of this property and translates the results into explicit descriptions of gapsets for Q_{{p,q}} and consecutive distances in ⟨p,q⟩. No equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the argument chain is presented as independent of prior fitted values or external uniqueness theorems from the same authors. The derivation therefore remains self-contained against the stated combinatorial inputs.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
-
IndisputableMonolith/Foundation/ArithmeticFromLogic.leanLogicNat recovery and embed_strictMono_of_one_lt unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Main Lemma: DΔ = {r_{i-1} − z r_i | 0 ≤ z ≤ Z_i − 1, 1 ≤ i ≤ t} via Euclidean algorithm on p and r1 (inverse of u).
-
IndisputableMonolith/Constants.leanphi_golden_ratio and phi_ladder constructions echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
Example 2 and Theorem 3: Fibonacci case p = F_k, q = F_{k+1} yields G = {F_m − 1}, with bounds F_k < p ≤ F_{k+1}.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
-
[1]
A. Al-Kateeb, M. Ambrosino, H. Hong, E. Lee,Maximum gap in cyclotomic polynomials, J. Number Theory 220 (2021), 1-15
work page 2021
-
[2]
Bachman,On ternary inclusion-exclusion polynomials, Integers 10 (2010), 623-638
G. Bachman,On ternary inclusion-exclusion polynomials, Integers 10 (2010), 623-638
work page 2010
- [3]
-
[4]
O. Camburu, E. Ciolan, F. Luca, P. Moree, I. Shparlinski,Cyclotomic coefficients: gaps and jumps, J. Number Theory 163 (2016), 211-237
work page 2016
-
[5]
H. Hong, E. Lee, H.-S Lee, C.-M. ParkMaximum gap in (inverse) cyclotomic polynomials, J. Number Theory 132 (10) (2012), 2297-2317
work page 2012
-
[6]
Moree,Numerical semigroups, cyclotomic polynomials, and Bernoulli numbers, Amer
P. Moree,Numerical semigroups, cyclotomic polynomials, and Bernoulli numbers, Amer. Math. Monthly 121 (10) (2014), 890-902
work page 2014
-
[7]
J.L. Ram´ ırez Alfonsin,The Diophantine Frobenius Problem, Oxford Lecture Series in Mathe- matics and its Applications, vol. 30, Oxford University Press, Oxford, 2005, 256pp
work page 2005
-
[8]
Sanna,A survey on coefficients of cyclotomic polynomials, Expo
C. Sanna,A survey on coefficients of cyclotomic polynomials, Expo. Math. 40 (3) (2022), 469-494
work page 2022
-
[9]
Zhang,Remarks on the maximum gap in binary cyclotomic polynomials, Bull
B. Zhang,Remarks on the maximum gap in binary cyclotomic polynomials, Bull. Math. Soc. Sci. Math. Roum. 59 (2016),109-115. Department of Mathematical Sciences, University of Nevada Las Vegas, 4505 Mary- land Parkway, Box 454020, Las Vegas, Nevada 89154-4020, USA Email address:gennady.bachman@unlv.edu
work page 2016
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.