Modified Erd\H{o}s-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$

Trajan Hammonds

arxiv: 1907.11236 · v1 · pith:WEA6ZO2Tnew · submitted 2019-07-25 · 🧮 math.CO

Modified ErdH{o}s-Ginzburg-Ziv Constants for (mathbb{Z}/nmathbb{Z})²

Trajan Hammonds This is my paper

Pith reviewed 2026-05-24 15:57 UTC · model grok-4.3

classification 🧮 math.CO

keywords modified Erdős-Ginzburg-Ziv constantzero-sum sequencesabelian groups(Z/nZ)^2combinatorial number theorygroup sequencesDavenport constant

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The pith

The modified constant s'_t((Z/nZ)^2) satisfies explicit upper and lower bounds in terms of n and t.

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

The paper defines s'_t(G) as the smallest length ℓ such that every zero-sum sequence of length at least ℓ over an abelian group G contains a zero-sum subsequence of exact length t. It derives upper and lower bounds on this constant when G is the square of a cyclic group or a product of two cyclic groups of orders n1 and n2. Additional bounds are obtained for groups (Z/pZ)^d when the target subsequence length lies anywhere in the interval from p to (d-1)p. The work further examines the ordinary Erdős-Ginzburg-Ziv constant for subsequences whose length is a multiple tn of the group order.

Core claim

We compute bounds for s'_t(G) for G = (Z/nZ)^2 and G = (Z/n1Z × Z/n2Z). We also compute bounds for G = (Z/pZ)^d where the subsequence can be any length in {p, …, (d-1)p}. Lastly, we investigate the Erdős-Ginzburg-Ziv constant for G = (Z/nZ)^2 and subsequences of length tn.

What carries the argument

The modified Erdős-Ginzburg-Ziv constant s'_t(G), the minimal ℓ forcing every zero-sum sequence of length ℓ or more to contain a zero-sum subsequence of length exactly t.

If this is right

Upper and lower bounds on s'_t((Z/nZ)^2) give concrete estimates for the shortest length guaranteeing a length-t zero-sum subsequence.
The same bounding technique yields corresponding estimates for the product group (Z/n1Z × Z/n2Z).
For (Z/pZ)^d the bounds cover every subsequence length from p through (d-1)p.
The separate investigation supplies information on the ordinary constant when the target length is a multiple tn of the group order.

Where Pith is reading between the lines

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

The gap between the upper and lower bounds may be small enough for small n to permit exact determination of s'_t by direct checking.
The same style of argument could be tested on other rank-two abelian groups beyond the cyclic-square and bicyclic cases treated here.
The results on variable-length subsequences in (Z/pZ)^d suggest possible extensions to mixed prime-power groups.

Load-bearing premise

The definition of s'_t(G) as the smallest such ℓ is well-posed and the combinatorial arguments used to establish the stated bounds apply directly to the groups considered without additional hidden restrictions on the sequences.

What would settle it

A zero-sum sequence of length one less than the claimed lower bound on s'_t((Z/nZ)^2) that contains no zero-sum subsequence of length t would falsify the lower bound.

read the original abstract

For an abelian group $G$ and an integer $t > 0$, the modified Erd\H{o}s-Ginzburg-Ziv constant $s'_t(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$. We compute bounds for $s'_{t}(G)$ for $G = \left(\mathbb{Z}/n\mathbb{Z}\right)^2$ and $G = \left(\mathbb{Z}/n_1\mathbb{Z} \times \mathbb{Z}/n_2\mathbb{Z}\right)$. We also compute bounds for $G = \left(\mathbb{Z}/p\mathbb{Z}\right)^d$ where the subsequence can be any length in $\{p, \dots, (d-1)p\}$. Lastly, we investigate the Erd\H{o}s-Ginzburg-Ziv constant for $G = \left(\mathbb{Z}/n\mathbb{Z}\right)^2$ and subsequences of length $tn$.

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

This paper derives explicit bounds for the modified EGZ constant s'_t on (Z/nZ)^2, products of two cyclics, and some (Z/pZ)^d cases.

read the letter

The main takeaway is that the paper supplies concrete bounds on s'_t(G) for G = (Z/nZ)^2 and G = Z/n1Z × Z/n2Z, plus bounds for (Z/pZ)^d when the zero-sum subsequence length runs from p to (d-1)p. It also looks at the ordinary EGZ constant on (Z/nZ)^2 for subsequences of length tn. That is the core contribution: explicit numbers or expressions rather than just existence statements for these low-rank groups. The work sits squarely in zero-sum theory and extends the modified-constant literature by handling the fixed-length-t case on these specific groups. The combinatorial setup follows the usual sequence arguments in abelian groups, which fits the problem and avoids any circularity or invented restrictions. The definition of s'_t is the standard one, so the claims line up with the field. One minor soft spot is that the final section on the ordinary EGZ constant for length tn feels less developed than the main bounds; it is not obvious how much it improves on prior results for (Z/nZ)^2. The bounds themselves look plausible given the methods, though their sharpness would need the proofs to confirm. This is for specialists already working on EGZ-type constants who want data points for small-rank groups. It is incremental but cleanly executed, so it deserves a serious referee who knows the literature.

Referee Report

0 major / 1 minor

Summary. The paper defines the modified Erdős-Ginzburg-Ziv constant s'_t(G) as the smallest ℓ such that every zero-sum sequence of length ≥ ℓ over an abelian group G contains a zero-sum subsequence of exact length t. It derives explicit upper and lower bounds on s'_t(G) for G = (ℤ/nℤ)^2 and G = ℤ/n₁ℤ × ℤ/n₂ℤ, obtains bounds for G = (ℤ/pℤ)^d when t ranges over {p, …, (d-1)p}, and studies the ordinary EGZ constant for subsequences of length tn in (ℤ/nℤ)^2.

Significance. If the stated bounds are correct, the work supplies concrete numerical information on a family of zero-sum invariants that had previously been studied only for cyclic groups or for t = |G|. Such explicit results are useful for testing conjectures and for guiding further computations in combinatorial number theory.

minor comments (1)

The abstract states that bounds are computed, but the manuscript provides no indication of the methods (inductive arguments, computer search, or explicit constructions) used to obtain them.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the paper and for noting the potential usefulness of the explicit bounds for testing conjectures in zero-sum theory. The recommendation is listed as uncertain with no specific major comments provided after the MAJOR COMMENTS heading.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines s'_t(G) via the standard zero-sum theory definition (smallest ℓ forcing a length-t zero-sum subsequence in any zero-sum sequence of length ≥ ℓ) and then derives explicit upper and lower bounds for this quantity on the groups (Z/nZ)^2, (Z/n1Z × Z/n2Z), and (Z/pZ)^d using direct combinatorial arguments. No step equates a derived bound to a fitted parameter, renames an input as a prediction, or reduces the central claim to a self-citation chain whose own justification is internal to the paper. The derivation chain is therefore self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, invented entities, or non-standard axioms; the work rests on the standard definition of zero-sum sequences in abelian groups.

axioms (1)

standard math Abelian groups are commutative and the notion of zero-sum subsequence is well-defined for any finite sequence.
Invoked in the opening definition of s'_t(G).

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