Size of the largest sum-free subset of $[n]^3$ and $[n]^4$

TL;DR

The paper resolves the asymptotic density of the largest sum-free subset of [n]^d for d=3 and d=4 by developing a linear programming relaxation and a dual weight-function framework that reduces upper bounds to constructing precise couplings on carefully partitioned regions of the unit cube. Through joint mixability and a discretization scheme, the authors build a weight function that enforces sum-free constraints with controlled error, showing |S_{d,n}| = c_d^* n^d + O(n^{d-1/2}) and matching the continuous optimum c_d^* for d=3,4. This yields c_d = c_d^* in these dimensions and provides a modular approach that could extend to higher dimensions given suitable couplings and simplicial decompositions. The work also contributes a general methodology linking probabilistic couplings, polytope geometry, and discrete-geometry bounds to settle long-standing questions in additive combinatorics.

Abstract

We determine the density of the largest sum-free subset of the lattice cube $\{1, 2, \dots, n\}^d$ for $d = 3$ and $d = 4$. This solves a conjecture of Cameron and Aydinian in dimensions $3$ and $4$.

Theorems & Definitions (50)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.2: Lower Bound
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5: Existence of Weight Function
• proof : Proof of \ref{['thm:main']}.