REVIEW 1 minor
Modified Erd\H{o}s-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$
T0 review · 0 major / 1 minor · reviewed 2026-05-24 · grok-4.3
Pith's one-line read The modified constant s'_t((Z/nZ)^2) satisfies explicit upper and lower bounds in terms of n and t.
desk verdict 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 pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
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.
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.
Extended reading notes
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.
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.
Editorial extensions
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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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
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
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.
Assumptions & free parameters
assumptions (1)
- standard math Abelian groups are commutative and the notion of zero-sum subsequence is well-defined for any finite sequence.
Cite this review
Pith. "Pith review of Modified Erd\H{o}s-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$." pith.science (2026). https://pith.science/paper/WEA6ZO2T
@misc{pith2026190711236,
author = {Pith},
title = {Pith review of: Modified Erd\Hos-Ginzburg-Ziv Constants for $(\mathbbZ/n\mathbbZ)^2$},
year = {2026},
howpublished = {\url{https://pith.science/paper/WEA6ZO2T}},
note = {Machine review of arXiv:1907.11236}
}
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 $tn$.
Reviewed May 24, 2026 · model on record in the stance chip above.
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