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arxiv: 2605.00816 · v2 · pith:ID72A2MUnew · 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 2026-07-01 07:36 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords sum-free subsetslattice cubehypercubelimiting densitylinear mapmeasure maximizationhyperplane slicesconvex bodies
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The pith

The largest sum-free subset of the lattice cube {1,2,…,n}^d has limiting density equal to the maximum measure of a sum-free set in the unit hypercube (0,1)^d, attained by a linear slab.

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

The paper determines the limiting density of the largest sum-free subset of the d-dimensional lattice cube for every fixed d. It proves this density equals the largest possible measure of a sum-free subset inside the continuous unit hypercube, and that this maximum is achieved precisely by sets of the form {x in (0,1)^d : 1 ≤ L(x) < 2} for a suitable linear map L. The result confirms the conjecture that the optimal construction arises from two parallel hyperplane slices. The authors also exhibit a convex body in high dimension where no such linear slab achieves the maximum measure.

Core claim

We show that the largest measure of a sum-free subset of the hypercube (0,1)^d is attained by the set {x ∈ (0,1)^d : 1 ≤ L(x) < 2} for some linear map L : R^d → R. Equivalently, the limiting density of the largest sum-free subset of {1,2,…,n}^d is determined by this construction.

What carries the argument

The linear slab {x : 1 ≤ L(x) < 2} for a linear map L : R^d → R, which is the portion of the hypercube lying between two parallel hyperplanes distance 1 apart in the L-direction.

If this is right

  • The limiting density for the lattice cube equals the measure of the optimal continuous linear slab.
  • The optimal discrete constructions are given asymptotically by lattice points inside such slabs.
  • The linear-slab construction fails to be optimal inside arbitrary convex bodies once the dimension is large enough.

Where Pith is reading between the lines

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

  • The counterexample for general convex bodies shows that the product structure of the cube is essential for the linear slab to be optimal.
  • Similar extremal problems on other domains such as the torus or balls may require different constructions.
  • Small-d computations of the exact maximum measure could give explicit values for the limiting density.

Load-bearing premise

The limiting density of the largest sum-free subset in the discrete lattice cube equals the maximum measure of a sum-free subset in the continuous unit hypercube.

What would settle it

A sum-free subset of (0,1)^d whose measure strictly exceeds that of every set {x : 1 ≤ L(x) < 2} for linear L, or a sequence of sum-free subsets of the lattice cube whose densities fail to approach the continuous maximum.

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. Equivalently, we show that the largest measure of a sum-free subset of the hypercube $(0,1)^d\subset \mathbb{R}^d$ is attained by $\setcond{x\in (0,1)^d}{1\leq L(x)<2}$ for some linear map $L:\mathbb{R}^d\to \mathbb{R}$. It is natural to conjecture that the same phenomenon might hold if one replaces the hypercube by any convex set not containing the origin, but we give an example to show that for sufficiently large $d$ this is not the case.

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

0 major / 1 minor

Summary. The manuscript determines the limiting density (as n→∞) of the largest sum-free subset of the d-dimensional lattice cube {1,…,n}^d for every fixed d, confirming that the extremal construction consists of two appropriate hyperplane slices. Equivalently, it proves that the maximum Lebesgue measure of a sum-free subset of the unit hypercube (0,1)^d is attained precisely by a set of the form {x : 1 ≤ L(x) < 2} for a suitable linear map L : R^d → R. The paper also supplies a counterexample showing that the same linear-slice phenomenon fails for sufficiently high-dimensional convex bodies not containing the origin.

Significance. If the central claims hold, the work exactly resolves a natural conjecture in additive combinatorics by replacing asymptotic bounds with a precise determination of the density for all d. The continuous formulation on the hypercube and the explicit counterexample for general convex sets clarify the boundary of the result and may inform related questions in discrete geometry.

minor comments (1)
  1. [Abstract] The abstract states the result cleanly; a brief parenthetical remark on the range of d (fixed versus growing with n) would help readers immediately see the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, which accurately summarizes the main results of the paper, and for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper resolves a pre-existing natural conjecture on the limiting density of sum-free subsets of the lattice cube by proving an equivalence to a continuous measure maximization problem on the hypercube, where the extremal sets are linear slices of the form {x : 1 ≤ L(x) < 2}. This is presented as a determination of a conjectured quantity rather than a self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The additional counterexample for general convex bodies demonstrates independent content outside any internal reduction. No quoted steps reduce the claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract alone; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned.

axioms (1)
  • standard math Basic definitions and properties of sum-free sets and Lebesgue measure on Euclidean space
    The equivalence between discrete and continuous formulations implicitly relies on standard facts from additive combinatorics and measure theory.

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10 extracted references · 1 canonical work pages

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