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arxiv: 2606.27961 · v1 · pith:DJTVEWT2new · submitted 2026-06-26 · 🧮 math.NT · cs.CR· cs.DM· math.CO

Transversal Difference Numbers in Finite Abelian Quotients

Mugurel Barcau , Vicen\c{t}iu Pa\c{s}ol , George C. \c{T}urca\c{s} This is my paper

Pith reviewed 2026-06-29 02:50 UTC · model grok-4.3

classification 🧮 math.NT cs.CRcs.DMmath.CO
keywords transversal difference numberfinite abelian groupsdifference setstransversalsKneser's theoremfinite fieldsGalois labels
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The pith

For G=(Z/p²Z)² and H=pG with odd prime p, the transversal difference number δ(G,H) is at least 3p²-p-1 and conjecturally equals (2p-1)².

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

The paper defines the transversal difference number δ(G,H) as the minimal size of the difference set D(T) over all transversals T of the quotient G/H in a finite abelian group G containing subgroup H. It proves the general lower bound δ(G,H) ≥ 2|G/H| - m(G,H), where m(G,H) is the largest order of a subgroup of G disjoint from H, and shows the bound is sharp for cyclic quotients while using Kneser's theorem to obtain exact values in certain product families. The core technical contribution isolates the square-plane case G=(Z/p²Z)², H=pG for odd primes p, where transversals are graphs of functions F_p² to F_p² and D(T) decomposes into carry-corrected finite-field derivative images; here the authors conjecture the exact value (2p-1)², prove the unconditional lower bound 3p²-p-1, and supply small-prime, probabilistic, and fixed-polynomial evidence. A sympathetic reader cares because this invariant connects factorization, tiling complements, and small-sumset problems, and arises in ambient Galois labels for cyclotomic-subfield homomorphic encryption.

Core claim

Transversals T for the quotient in the square-plane case are graphs of functions from F_p² to F_p², and their difference supports D(T) decompose into carry-corrected finite-field derivative images. The authors conjecture that δ(G,H)=(2p-1)² for all odd primes p, prove the unconditional lower bound δ(G,H)≥3p²-p-1, and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.

What carries the argument

Decomposition of D(T) into carry-corrected finite-field derivative images for graph transversals from F_p² to F_p²

If this is right

  • The general lower bound is achieved precisely when the quotient is cyclic.
  • Kneser's theorem produces exact product families when one coordinate is nonsplit cyclic and the remaining factors are arbitrary split.
  • The square-plane case is the first genuinely new residual obstruction beyond the general bound.
  • If the conjecture holds, then δ(G,H) equals (2p-1)² exactly for every odd prime p.

Where Pith is reading between the lines

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

  • The same decomposition technique could be tested on higher p-powers or other non-cyclic p-adic groups to seek closed-form exact values.
  • The connection to small-sumset questions suggests possible direct comparisons with Cauchy-Davenport bounds over finite fields.
  • Systematic computation for the next few primes beyond the small cases already checked would provide a direct test of the conjecture.

Load-bearing premise

The decomposition of D(T) into carry-corrected finite-field derivative images for transversals that are graphs of functions F_p² → F_p² fully captures the minimal size without additional hidden relations or overcounting that would invalidate the conjectured exact value.

What would settle it

An explicit transversal T for some odd prime p with |D(T)| smaller than (2p-1)² would falsify the conjecture, while a construction with |D(T)| smaller than 3p²-p-1 would contradict the proven lower bound.

read the original abstract

Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\). We call $ \delta(G,H)=\min_T |D(T)| $ the transversal difference number of the pair \((G,H)\). This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound $\delta(G,H)\geq 2|G/H|-m(G,H), $ where \(m(G,H)\) is the largest order of a subgroup of \(G\) disjoint from \(H\). The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane \[ G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. \] For odd \(p\), this case is the technical core of the paper. Here transversals are graphs of functions \(\mathbb F_p^2\to \mathbb F_p^2\), and \(D(T)\) decomposes into carry-corrected finite-field derivative images. We conjecture that \[ \delta(G,H)=(2p-1)^2 \] for all odd primes \(p\), prove the unconditional lower bound \(3p^2-p-1\), and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.

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

1 major / 2 minor

Summary. The manuscript defines the transversal difference number δ(G,H) as the minimum |D(T)| over transversals T of G/H in finite abelian groups G ⊃ H. It proves the general lower bound δ(G,H) ≥ 2|G/H| − m(G,H) with m(G,H) the largest order of a subgroup of G disjoint from H; shows sharpness for cyclic quotients; obtains exact values for certain product families via Kneser’s theorem; and, for the nonsplit square-plane case G = (ℤ/p²ℤ)², H = pG with odd prime p, conjectures δ(G,H) = (2p−1)², proves the unconditional lower bound δ(G,H) ≥ 3p² − p − 1, and supplies small-prime computational, probabilistic, and fixed-polynomial evidence.

Significance. The general lower bound and its sharpness for cyclic quotients are cleanly established and constitute a solid contribution to difference-set problems in abelian groups. The identification of the square-plane case as the first genuinely new obstruction, together with the explicit modeling via carry-corrected derivatives on graph transversals, isolates a concrete open question whose resolution would have implications for finite abelian factorizations and tiling complements. The computational evidence for small p is a positive feature, though the exact conjecture remains open.

major comments (1)
  1. [square-plane case section] Square-plane case (abstract and dedicated section): the conjecture δ(G,H) = (2p−1)² is supported by the decomposition of D(T) into carry-corrected finite-field derivative images for graph transversals, yet the manuscript only proves the strictly weaker unconditional bound 3p² − p − 1; it is not shown that the graph representation and carry correction are exhaustive, so hidden additive relations in ℤ/p²ℤ could in principle permit |D(T)| between 3p² − p − 1 and (2p−1)².
minor comments (2)
  1. [evidence subsection] The phrase 'fixed-polynomial evidence' in the abstract should be expanded in the evidence subsection to specify the precise polynomial family and the range of p for which it was verified.
  2. [square-plane case section] Notation for the carry-correction map and the finite-field derivative should be introduced with a displayed equation before its first use in the square-plane analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. The single major comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [square-plane case section] Square-plane case (abstract and dedicated section): the conjecture δ(G,H) = (2p−1)² is supported by the decomposition of D(T) into carry-corrected finite-field derivative images for graph transversals, yet the manuscript only proves the strictly weaker unconditional bound 3p² − p − 1; it is not shown that the graph representation and carry correction are exhaustive, so hidden additive relations in ℤ/p²ℤ could in principle permit |D(T)| between 3p² − p − 1 and (2p−1)².

    Authors: We agree that the manuscript does not contain an explicit lemma proving exhaustiveness of the graph parameterization. Every transversal T for G/H can nevertheless be represented as the graph of a function f: F_p² → F_p² once a fixed section of the quotient map is chosen; the difference set D(T) is then completely determined by the carry-corrected derivatives of f. This follows directly from the coordinate-wise addition law in (ℤ/p²ℤ)², where all carries are explicitly recorded in the model and no further additive relations remain. The unconditional lower bound 3p²−p−1 is derived independently of this parameterization. To remove any ambiguity we will insert a short clarifying lemma establishing that the representation is exhaustive and that the conjecture is therefore posed over the complete set of transversals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the new invariant δ(G,H) explicitly as min_T |D(T)| and derives the general lower bound δ(G,H) ≥ 2|G/H| - m(G,H) from subgroup order considerations and Kneser's theorem without reference to the specific (G,H) conjecture. For the square-plane case the conjecture δ(G,H)=(2p-1)^2 is stated as a separate claim, supported by a strictly weaker unconditional bound 3p²-p-1 that is proved independently; the graph-of-functions representation and carry-corrected decomposition are presented as a modeling choice that generates the conjectured value rather than defining it by construction. No load-bearing self-citations, fitted inputs renamed as predictions, or ansätze smuggled via prior work appear in the provided text. The central claims therefore remain independent of their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from finite abelian group theory and Kneser's theorem; no free parameters or invented entities are introduced in the abstract. The conjecture itself is not derived from additional axioms beyond the group structure.

axioms (1)
  • standard math Kneser's theorem on sumsets in abelian groups
    Invoked to obtain exact product families with one nonsplit cyclic coordinate.

pith-pipeline@v0.9.1-grok · 5884 in / 1378 out tokens · 31583 ms · 2026-06-29T02:50:53.931482+00:00 · methodology

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Reference graph

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