On the largest sum-free subset of the lattice cube

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

• The paper proves that the maximal sum-free subset of {1,…,n}^d is asymptotically c_d* n^d, achieved via a double-slice structure based on coordinate sums.
• It employs a novel linear programming duality and measure-theoretic induction to overcome joint mixability barriers in high-dimensional lattice cubes.
• Sharp volume estimates using large deviation principles and log-concavity confirm the optimality of the double-slice configuration and highlight contrasts with general convex bodies.

Sum-Free Subsets in Lattice Cubes: Asymptotic Density and Structure

Introduction and Main Result

The determination of the maximum density of sum-free subsets in high-dimensional lattice cubes is a classical problem in additive combinatorics. A set $S\subset\{1,2,\ldots,n\}^d$ is sum-free if there are no $x,y,z\in S$ with $x+y=z$, with vector addition. While the one-dimensional case is trivial ($c_1=1/2$), and closed solutions for $d=2,3,4$ have appeared in recent years, the case of general $d$ has remained open. This paper proves that, for all fixed $d\geq 1$, the size of the largest sum-free subset of $\{1,2,\ldots,n\}^d$ is asymptotically $c_d^* n^d$ as $n\to\infty$, where $x,y,z\in S$0 is the measure of a maximal "double-slice" constructed via sums of coordinates in the continuous cube $x,y,z\in S$1.

Specifically, for appropriate $x,y,z\in S$2, the asymptotically extremal sum-free subset is

$x,y,z\in S$3

and $x,y,z\in S$4. The conjecture that this double-slice structure is optimal for all $x,y,z\in S$5 is resolved affirmatively in this work.

Technical Approach and Joint Mixability

The proof builds on the framework of Lepsveridze and Sun for $x,y,z\in S$6 (Lepsveridze et al., 2023), generalising their methods. A crucial ingredient is the translation of the sum-free problem to an associated linear program (LP) over the collection of 1-dimensional fibers determined by projections onto lower dimensional subcubes. The LP's dual involves assigning weights to triples $x,y,z\in S$7 with $x,y,z\in S$8 to bound the total possible fiber weight.

A central and previously unresolved obstacle—joint mixability of certain uniform measures on slices of the cube—is resolved here for all $x,y,z\in S$9. The authors prove (Theorem~\ref{thm:joint-mix}) that for $x+y=z$0 and any nonnegative integers $x+y=z$1 with $x+y=z$2, the product measures $x+y=z$3, $x+y=z$4, $x+y=z$5 are jointly mixable: there exists a coupling $x+y=z$6 with $x+y=z$7 and $x+y=z$8. This is equivalent to finding uniform probability couplings on corresponding slices that add up coordinatewise to a fixed vector, generalising previous constructive or case-by-case approaches to a fully general setting.

The method employs a sophisticated measure-theoretic induction, relying on Markov chain ergodicity and fixed-point theorems. This not only answers a structural question about high-dimensional coupling but underpins the feasibility of optimal dual solutions for the sum-free LP.

Sharp Slice Estimates and Weight Function Construction

Through careful analysis of the continuous cube and sums of independent uniform random variables, the paper derives sharp asymptotics for the relevant volumes of slices (Lemma~\ref{lem:real-layer-estimate}). Using large deviation principles, convexity, and log-concavity, the authors show that $x+y=z$9 is asymptotically $c_1=1/2$0, with the slice densities decaying at quantifiable exponential rates off the peak. These slice volume estimates are crucial: they ensure that mass can be distributed so that sum-free constraints are satisfied to leading order, and all dual constraints in the LP formulation can be met.

Using the above, the authors explicitly construct a feasible set of weights $c_1=1/2$1 for the dual LP. These weights are supported on certain triples of fibers corresponding to slices whose sums are in feasible sum-free ranges, and are calibrated using the proven mixability results and slice estimates. The construction is delicate: it partitions the cube into regions indexed by sums of coordinates and assigns weights that ensure the sum-free constraints are met up to lower order terms.

Extensions, Contrasts, and Open Questions

This work further explores whether analogous extremal structures hold for sum-free subsets in general convex bodies, not just cubes. While the double-slice construction is optimal for the cube, the paper demonstrates—by explicit examples and volume estimates—that this is not universally true for arbitrary convex sets in sufficiently high dimensions. For some convex bodies, there exists a sum-free set that has larger measure than any slice defined by a linear functional modulo 3. However, the authors conjecture that for convex subsets of $c_1=1/2$2 the slice construction might still be optimal, leaving open the complete characterisation of extremal sum-free subsets in low-dimensional convex domains.

Numerical and Algorithmic Verification

Theoretical arguments are supplemented by verified bounds for small dimensions using recurrence relations for slice volumes and explicit computations (see the appendix and [J13]). Code for these numerical checks is provided, which validates the sharpness of the volume ratios used in the weight construction up to $c_1=1/2$3.

Implications and Future Directions

The resolution of the asymptotic density and structure of maximal sum-free subsets in high-dimensional lattice cubes sharpens the understanding of additive structure in product spaces. The general joint mixability result closes a technical barrier in understanding multi-marginal couplings and can impact the analysis of combinatorial designs, probability, and applications to theoretical computer science, such as random generation of constrained objects.

From a broader perspective, this work exemplifies the synthesis of analytic, probabilistic, and combinatorial methods to resolve extremal questions in discrete geometry. The negative answer for general convex bodies in high dimensions suggests new directions in geometric additive combinatorics, inviting characterisation of exceptional shapes and potential symmetry-breaking phenomena.

Conclusion

The paper settles a fundamental open problem on the maximal sum-free density in lattice cubes of arbitrary fixed dimension, confirming that two hyperplane slices optimally describe the extremal configuration. The introduction and proof of a strong joint mixability property in discrete cubes is a robust tool likely to find utility in various areas requiring explicit high-dimensional couplings. The contrast observed for general convex sets hints at further intricate combinatorial geometry awaiting exploration.

Reference: "On the largest sum-free subset of the lattice cube" (2605.00816).