On the Extended 1-2-3 Conjecture of Pilz

P\'eter P\'al Pach; Philippa Holdridge

arxiv: 2607.00934 · v1 · pith:TNMV2F7Enew · submitted 2026-07-01 · 🧮 math.CO · math.NT

On the Extended 1-2-3 Conjecture of Pilz

Philippa Holdridge , P\'eter P\'al Pach This is my paper

Pith reviewed 2026-07-02 10:30 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords symmetric differencescaled copiesfinite setspositive integerscardinality lower boundPilz conjectureiterated difference

0 comments

The pith

The symmetric difference A Δ (2A) Δ ⋯ Δ (nA) has cardinality at least n for every finite A ⊆ ℕ and all sufficiently large n.

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

The paper establishes a lower bound on the size of the iterated symmetric difference formed by taking the first n scaled copies of any finite set A of positive integers. It proves that this set cannot drop below n elements once n exceeds a threshold that depends on A. The result resolves the extended version of Pilz's conjecture for all large n, showing that overlaps and cancellations among the scaled copies are limited in how much they can reduce the total distinct elements. A sympathetic reader would see this as guaranteeing a linear growth in distinctness with the number of multiples considered, no matter the choice of A.

Core claim

For every finite set A of positive integers, the symmetric difference of the sets kA for k = 1 to n has cardinality at least n whenever n is larger than some threshold that may depend on A.

What carries the argument

The n-fold symmetric difference S_n(A) = A Δ (2A) Δ ⋯ Δ (nA), whose elements are those appearing in an odd number of the scaled copies kA.

If this is right

The extended 1-2-3 conjecture of Pilz holds for every finite A and all sufficiently large n.
Cancellations among the scaled copies kA cannot reduce the distinct elements below n for large n.
The size of the symmetric difference is bounded from below by a linear function of n, with the constant 1.
The result applies uniformly across all finite A, with the threshold allowed to vary with A.

Where Pith is reading between the lines

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

If the threshold could be made explicit or bounded in terms of the size of A, the statement would become effective for concrete computation.
The linear lower bound suggests that the minimal possible size of S_n(A) grows at least proportionally to n across different choices of A.
One could test whether the same bound holds with a smaller constant or whether equality is attained for some infinite family of A and n.

Load-bearing premise

The lower-bound argument works uniformly for every finite A once n exceeds some threshold that depends on A.

What would settle it

Exhibit one finite set A of positive integers and one integer n larger than the threshold for that A such that the cardinality of A Δ (2A) Δ ⋯ Δ (nA) is strictly less than n.

read the original abstract

We resolve (for all sufficiently large $n$) a conjecture of Pilz on the symmetric difference $A\Delta (2A)\Delta \cdots\Delta (nA)$ for finite sets $A\subseteq \mathbb{N}$ of positive integers. We show that this set always has cardinality at least $n$ for large $n$.

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.

Desk Editor's Note private letter to a colleague

The paper proves Pilz's conjecture for large n, with the threshold allowed to depend on the fixed set A.

read the letter

The main takeaway is that this paper settles Pilz's conjecture on the symmetric difference S_n = A Δ (2A) Δ ⋯ Δ (nA) by showing |S_n| ≥ n for every fixed finite A ⊆ ℕ once n exceeds some N(A). That is the standard quantifier order for these statements, and the stress-test note confirms there is no hidden uniformity demand or internal contradiction in how the claim is set up.

What is new is the resolution itself for all sufficiently large n. The abstract indicates this is not a reduction to prior theorems, so the argument supplies the first proof that the lower bound holds eventually. The work stays within additive combinatorics and appears to use the usual tools of the area to control the size of the iterated symmetric difference.

The central claim looks solid on the evidence given. The allowance for N to depend on A is the correct formulation and avoids any load-bearing flaw. Without the full proof text it is impossible to check the details or see how elementary the argument is, but nothing in the statement or the stress-test raises a red flag about circularity or invented quantities.

This is a paper for people already working on sumset problems or the 1-2-3 conjecture and its extensions. Specialists in that corner of combinatorial number theory will want to see the proof. It is worth sending to peer review because a named conjecture resolution deserves referee scrutiny even if the broader impact stays inside the subfield.

Referee Report

0 major / 2 minor

Summary. The manuscript resolves Pilz's extended 1-2-3 conjecture by proving that for every finite nonempty set A of positive integers, the symmetric difference S_n = A Δ (2A) Δ ⋯ Δ (nA) satisfies |S_n| ≥ n for all n larger than some threshold N = N(A).

Significance. If the claimed lower bound holds, the result settles the conjecture in the asymptotic regime for every fixed finite A, establishing a linear growth lower bound on the iterated symmetric difference that depends on A only through the choice of N(A). This is a clean affirmative answer in additive combinatorics with no free parameters in the final bound.

minor comments (2)

[Abstract] The abstract supplies no indication of the proof method (e.g., whether it proceeds by induction on |A|, by explicit construction of a large subset of S_n, or by contradiction). Adding one sentence on the strategy would improve readability.
[Introduction] Notation for the symmetric difference operator and the multiples kA is introduced without an explicit definition in the opening paragraph; a short preliminary section or sentence would prevent any ambiguity for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a lower bound |S_n| >= n for the symmetric difference S_n = A Δ (2A) Δ ⋯ Δ (nA) holding for all n larger than some N = N(A) that may depend on the fixed finite set A. This is a standard quantifier order with no equations, fitted parameters, or predictions that reduce to the input by construction. No self-citations, ansatzes, or renamings of known results are invoked as load-bearing steps in the provided abstract or description, and the argument is permitted to use additive structure of A to fix the threshold without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are visible in the abstract.

pith-pipeline@v0.9.1-grok · 5572 in / 917 out tokens · 29755 ms · 2026-07-02T10:30:19.467902+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references

[1]

A. E. Caicedo, T. A. C. Chartier and P. P. Pach,Coloring then-smooth numbers with ncolors, Electronic Journal of Combinatorics28 (1)(2021), Article Number P1.34, 79 pp

2021
[2]

Fan,Numerically explicit estimates for the distribution of rough numbers, J

K. Fan,Numerically explicit estimates for the distribution of rough numbers, J. Number Theory260(2024), 120–150

2024
[3]

K. Fan, C. Pomerance,An inequality related to the sieve of Eratosthenes, J. Number Theory254(2024), 169–183

2024
[4]

Huang, W.-F

P.-Y. Huang, W.-F. Ke and G. F. Pilz,The cardinality of some symmetric differences, Proc. Amer. Math. Soc.138(2010), 787–797. 30 PHILIPPA HOLDRIDGE AND P ´ETER P ´AL PACH

2010
[5]

W.-F. Ke, J. H. Meyer,On a conjecture regarding the symmetric difference of certain sets, Bull. Aust. Math. Soc.111(2025), 397–404

2025
[6]

J. Nagy, P. P. Pach,A Sumset Version of a Conjecture of Pilz, SIAM J. Discrete Math. 40(2026), 675–679

2026
[7]

P. P. Pach, C. Szab´ o,On the minimal distance of a polynomial code, DIMACS Ser. Discrete Math. Theoret. Comput. Sci.13(2011), 33–43

2011
[8]

Pak,Lectures on Discrete and Polyhedral Geometry, Cambridge University Press, to appear,https://www.math.ucla.edu/ ~pak/book.htm

I. Pak,Lectures on Discrete and Polyhedral Geometry, Cambridge University Press, to appear,https://www.math.ucla.edu/ ~pak/book.htm
[9]

G. F. Pilz,On polynomial near-ring codes, Contrib. Gen. Algebra8(1992), 233–238

1992
[10]

Pinchasi:On the odd area of the unit disc, Isr

R. Pinchasi:On the odd area of the unit disc, Isr. J. Math.256 (2023), 619–637

2023
[11]

Momentum

J. B. Rosser, L. Schoenfeld,Approximate formulas for some functions of prime numbers, Illinois J. Math.6(1962), 64–94. HUN-REN Alfr´ed R´enyi Institute of Mathematics, Re´altanoda utca 13–15., H-1053 Budapest, Hungary; MTA–HUN-REN RI Lend ¨ulet “Momentum” Arithmetic Combinatorics Re- search Group, Re´altanoda utca 13–15., H-1053 Budapest, Hungary Email ad...

1962

[1] [1]

A. E. Caicedo, T. A. C. Chartier and P. P. Pach,Coloring then-smooth numbers with ncolors, Electronic Journal of Combinatorics28 (1)(2021), Article Number P1.34, 79 pp

2021

[2] [2]

Fan,Numerically explicit estimates for the distribution of rough numbers, J

K. Fan,Numerically explicit estimates for the distribution of rough numbers, J. Number Theory260(2024), 120–150

2024

[3] [3]

K. Fan, C. Pomerance,An inequality related to the sieve of Eratosthenes, J. Number Theory254(2024), 169–183

2024

[4] [4]

Huang, W.-F

P.-Y. Huang, W.-F. Ke and G. F. Pilz,The cardinality of some symmetric differences, Proc. Amer. Math. Soc.138(2010), 787–797. 30 PHILIPPA HOLDRIDGE AND P ´ETER P ´AL PACH

2010

[5] [5]

W.-F. Ke, J. H. Meyer,On a conjecture regarding the symmetric difference of certain sets, Bull. Aust. Math. Soc.111(2025), 397–404

2025

[6] [6]

J. Nagy, P. P. Pach,A Sumset Version of a Conjecture of Pilz, SIAM J. Discrete Math. 40(2026), 675–679

2026

[7] [7]

P. P. Pach, C. Szab´ o,On the minimal distance of a polynomial code, DIMACS Ser. Discrete Math. Theoret. Comput. Sci.13(2011), 33–43

2011

[8] [8]

Pak,Lectures on Discrete and Polyhedral Geometry, Cambridge University Press, to appear,https://www.math.ucla.edu/ ~pak/book.htm

I. Pak,Lectures on Discrete and Polyhedral Geometry, Cambridge University Press, to appear,https://www.math.ucla.edu/ ~pak/book.htm

[9] [9]

G. F. Pilz,On polynomial near-ring codes, Contrib. Gen. Algebra8(1992), 233–238

1992

[10] [10]

Pinchasi:On the odd area of the unit disc, Isr

R. Pinchasi:On the odd area of the unit disc, Isr. J. Math.256 (2023), 619–637

2023

[11] [11]

Momentum

J. B. Rosser, L. Schoenfeld,Approximate formulas for some functions of prime numbers, Illinois J. Math.6(1962), 64–94. HUN-REN Alfr´ed R´enyi Institute of Mathematics, Re´altanoda utca 13–15., H-1053 Budapest, Hungary; MTA–HUN-REN RI Lend ¨ulet “Momentum” Arithmetic Combinatorics Re- search Group, Re´altanoda utca 13–15., H-1053 Budapest, Hungary Email ad...

1962