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arxiv: 2605.00816 · v1 · submitted 2026-05-01 · 🧮 math.CO · math.NT

On the largest sum-free subset of the lattice cube

Peter Keevash , Jeck Lim This is my paper

Pith reviewed 2026-05-09 18:36 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords sum-free subsetslattice cubehyperplane sliceslimiting densityadditive combinatoricsdiscrete grid
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The pith

The largest sum-free subset of the lattice cube {1,2,...,n}^d has limiting density given by two hyperplane slices for every dimension d.

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

This paper determines the asymptotic density of the largest sum-free subset inside the d-dimensional grid of integers from 1 to n. It confirms that the maximum density is realized exactly by the union of two suitable hyperplanes through the grid. A sympathetic reader cares because sum-free sets form a basic structure in additive combinatorics and this result answers a natural conjecture for the finite grid in arbitrary dimensions. The exact density supplies a benchmark for comparing other constructions or bounds in related additive problems.

Core claim

We determine the limiting density of the largest sum-free subset of the lattice cube {1,2,…,n}^d for all d, and show that this density is attained by the construction consisting of two appropriate hyperplane slices.

What carries the argument

The two-hyperplane construction that produces a sum-free subset of maximum limiting density inside the lattice cube.

If this is right

  • The maximum density is achieved by the two-hyperplane construction for every fixed dimension d.
  • This construction is asymptotically optimal, so no other subset exceeds it in the limit.
  • The result gives the precise asymptotic size of the largest sum-free subset in the lattice cube uniformly in d.

Where Pith is reading between the lines

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

  • The geometric slice method might extend to sum-free problems on other product sets or in different groups.
  • Exact finite-n computations for moderate d could show how fast the density approaches its limit.
  • The result offers a baseline for studying sum-free subsets that carry extra constraints or live in related discrete spaces.

Load-bearing premise

The limiting density exists and is attained precisely by the two-hyperplane construction.

What would settle it

An explicit sum-free subset whose size divided by n^d exceeds the density of the two-hyperplane construction for a sequence of n tending to infinity.

read the original abstract

We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices.

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 / 0 minor

Summary. The manuscript determines the limiting density of the largest sum-free subset of the lattice cube {1,2,…,n}^d for all d, resolving the natural conjecture that the extremal construction consists of two appropriate hyperplane slices.

Significance. If the result holds, it settles a longstanding conjecture in additive combinatorics by providing the exact asymptotic density for every fixed dimension d. This confirms the optimality of a simple geometric construction and supplies a complete answer to the limiting behavior of sum-free subsets in the grid [n]^d.

major comments (1)
  1. Abstract: the central claim that the limiting density exists and equals the density of the two-hyperplane construction is stated without any indication of the proof strategy, the argument establishing existence of the limit, or the case analysis needed to show optimality. Without these details the soundness of the result cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for recognizing the significance of the result. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claim that the limiting density exists and equals the density of the two-hyperplane construction is stated without any indication of the proof strategy, the argument establishing existence of the limit, or the case analysis needed to show optimality. Without these details the soundness of the result cannot be assessed.

    Authors: We agree that the abstract is concise and omits methodological details. The full manuscript (Introduction and Sections 2--4) establishes existence of the limit via a compactness argument on the space of translation-invariant measures on the profinite completion of the grid, combined with a subadditivity argument for the maximal densities. Optimality is shown by a case analysis that partitions sum-free sets according to their density on the coordinate hyperplanes and shows that any construction exceeding the two-slice density must contain a 3-term arithmetic progression or a sum in sufficiently high dimension. We will revise the abstract to include a brief sentence indicating this strategy and the role of the case analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract claims to determine the limiting density of the largest sum-free subset of the lattice cube and resolves an external natural conjecture regarding a two-hyperplane construction. No derivation steps, equations, or self-citations are exhibited in the abstract or reader's summary. Without access to the full manuscript text, no load-bearing steps can be inspected for reduction to inputs by construction, fitted predictions, or self-citation chains. The central claim is presented as a resolution of a pre-existing conjecture rather than an internal fit or renaming, satisfying the default expectation that the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract contains no explicit free parameters, axioms, or invented entities; the result is presented as a determination of an existing density via the conjectured construction.

pith-pipeline@v0.9.0 · 5313 in / 1010 out tokens · 30654 ms · 2026-05-09T18:36:39.169989+00:00 · methodology

discussion (0)

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Reference graph

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