On the largest size of sum-free sets in symmetric regions

Abstract: A subset $S$ of a group $G$ is said to be sum-free (resp. $Δ$-free) if there are no solutions to $a+b=c$ (resp. $a+b+c=0$) with $a,b,c\in S$. For a convex region $R\subset\mathbb{R}<sup>d$, let $σ(R)$ denote the maximal proportion of the volume of $R$ that a sum-free subset of $R$ can occupy. We prove that $σ([-1,1]<sup>d)=1/2$. Our proof employs a careful application of the Brunn-Minkowski inequality. Moreover, for the $d$-dimensional Euclidean ball $\mathbb{B}<sup>d(0,1)$, we show that $σ(\mathbb{B}^d(0,1))\leq 1/2+o_d(1)$. We present two arguments for this. The first combines some routine harmonic analysis on the sphere with known bounds on values of the ultraspherical polynomials. The second more elementary argument proceeds by establishing that the maximal $Δ$-free subset of the unit sphere $\mathbb{S}<sup>{d-1}$ occupies $1/2+O(d<sup>{-1})$ of the sphere's surface measure. This answers a question raised by Bukh.

• The paper proves that the maximum density of a sum-free subset of the symmetric cube \([-1,1]^d\) is exactly \(1/2\) in every dimension using an orthant restriction and the Brunn–Minkowski inequality.
• The paper establishes that every \(\Delta\)-free subset of the sphere has density at most \(1/2+1/(2d)\) in odd dimensions and \(1/2+1/[2(d+1)]\) in even dimensions, resolving Bukh’s question asymptotically.
• The paper combines spherical harmonic analysis with unit-distance digraph methods to show that sum-free subsets of the Euclidean ball have density at most \(1/2+e^{-cd}\), while leaving the exact universal \(1/2\) bound open.

Overview

This paper studies extremal densities of sum-free and $\Delta$-free subsets of symmetric convex regions in $\mathbb{R}^d$. For a convex region $R \subset \mathbb{R}^d$, $\sigma(R)$ denotes the maximal proportion of the volume of $R$ that a sum-free subset may occupy; analogously, $\tau(R)$ denotes the maximal measure of a $\Delta$-free subset (one containing no solutions to $a+b+c=0$). The paper makes three principal contributions: an exact determination of $\sigma([-1,1]^d)$, a near-half asymptotic upper bound for $\sigma(\mathbb{B}^d(0,1))$ via harmonic analysis on the sphere, and a resolution of Bukh's question on $\mathbb{R}^d$0-free subsets of $\mathbb{R}^d$1 with an explicit error term.

Sum-free subsets of the cube

The main theorem establishes that $\mathbb{R}^d$2 for all $\mathbb{R}^d$3 (2607.07991). The lower bound is elementary: $\mathbb{R}^d$4 is sum-free. The upper bound is a short and sharp application of the Brunn-Minkowski inequality. Given a sum-free Borel set $\mathbb{R}^d$5, one selects an orthant $\mathbb{R}^d$6 with $\mathbb{R}^d$7 and applies Brunn-Minkowski to $\mathbb{R}^d$8 and $\mathbb{R}^d$9, obtaining $R \subset \mathbb{R}^d$0. Self-similarity of the cube gives $R \subset \mathbb{R}^d$1, so $R \subset \mathbb{R}^d$2 is disjoint from $R \subset \mathbb{R}^d$3, and the volume budget yields $R \subset \mathbb{R}^d$4.

The proof is notable for its economy. The authors observe that a naive Brunn-Minkowski argument applied directly to $R \subset \mathbb{R}^d$5 gives only the weak bound $R \subset \mathbb{R}^d$6; the orthant restriction is essential. The result places the symmetric cube alongside the unit cube $R \subset \mathbb{R}^d$7, for which $R \subset \mathbb{R}^d$8 was established for $R \subset \mathbb{R}^d$9 by Lepsveridze and Sun and for $\sigma(R)$0 by Keevash and Lim. However, the paper leaves open the natural generalization, stated as Conjecture: $\sigma(R)$1 for every symmetric convex region $\sigma(R)$2. Notably, the authors concede that they cannot prove this even for the Euclidean ball, and they present evidence that the conjecture, if true, would identify the cube as extremal among symmetric convex bodies. They also note that Erdős's middle-thirds construction gives only $\sigma(R)$3, so the lower bound may be far from the upper bound of $\sigma(R)$4.

$\sigma(R)$5-free sets under Gaussian measure and on the sphere

A key technical tool is a directed version of Mantel's theorem due to Brown and Harary, reproved in the paper: a digraph on $\sigma(R)$6 vertices with no directed triangle has at most $\sigma(R)$7 edges. This yields a novel proof of the known fact that $\sigma(R)$8 for the standard Gaussian measure $\sigma(R)$9 on $R$0: sampling $R$1 i.i.d. $R$2 vectors and joining $R$3 when $R$4 produces a directed-triangle-free digraph, and the Mantel-type bound forces $R$5 for all $R$6. While this result was previously derivable from Borell's Gaussian Brunn-Minkowski inequality, this probabilistic/digraph proof is what motivates the spherical argument.

The main result of this section answers a question of Bukh affirmatively in its asymptotic form:

$R$7

The proof is a discretization argument: the vertices of a regular unit simplex form a set $R$8 of size $R$9, and any $\tau(R)$0-free $\tau(R)$1 induces a directed-triangle-free digraph on $\tau(R)$2 vertices, so the Brown-Harary bound gives the stated density. Averaging over rotations transfers the bound to the whole sphere. The paper notes that Theorem was proved independently by Ducz and by Ge and Xu, with the present proof obtained in September 2025 and presented at IMPA in March 2026.

Connection to unit distance graphs

The paper develops a structural connection between $\tau(R)$3-free sets and triangle-free subgraphs of unit distance digraphs. Defining $\tau(R)$4 as the infimum, over finite unit distance digraphs in $\tau(R)$5, of the ratio $\tau(R)$6 (where $\tau(R)$7 is the maximum edge count of a directed-triangle-free subgraph), the paper proves $\tau(R)$8. This is proved by averaging the indicator of a $\tau(R)$9-free $\Delta$0 over random rotations applied to edge differences of $\Delta$1.

Two results frame the utility of this approach. On the upper side, the authors conjecture that $\Delta$2, which would substantially strengthen Theorem; they propose Hamming digraphs on $\Delta$3 (edges between points at Hamming distance $\Delta$4 for $\Delta$5) as candidate witnesses, and remark that a spectral proof of the undirected version should be feasible. On the lower side, the paper proves $\Delta$6 using the Frankl-Wilson / Larman-Rogers bounds $\Delta$7: a random bipartition of the $\Delta$8 color classes retains a $\Delta$9 fraction of edges in expectation, and bipartite digraphs have no directed triangles. This shows the graph-theoretic route cannot yield better than exponentially small error, so the exact constant $a+b+c=0$0 is out of reach by this method.

Harmonic-analytic upper bound for the ball

The second proof technique targets the ball via the sphere. For sum-free $a+b+c=0$1 with $a+b+c=0$2, the authors consider the cap $a+b+c=0$3; since $a+b+c=0$4 whenever $a+b+c=0$5, sum-freeness forces $a+b+c=0$6 for $a+b+c=0$7. Expanding $a+b+c=0$8 into spherical harmonics and evaluating $a+b+c=0$9 for $\sigma([-1,1]^d)$0 uniform on $\sigma([-1,1]^d)$1, one obtains the upper bound $\sigma([-1,1]^d)$2 from the cap constraint, while harmonic expansion gives $\sigma([-1,1]^d)$3. The crucial input is a bound of Castro-Silva, Filho, Slot, and Vallentin: $\sigma([-1,1]^d)$4 for all $\sigma([-1,1]^d)$5 and large $\sigma([-1,1]^d)$6. Combining with Parseval's identity yields $\sigma([-1,1]^d)$7, and integrating over spherical shells extends this to $\sigma([-1,1]^d)$8. The authors explicitly flag that this may be far from optimal given the $\sigma([-1,1]^d)$9 lower bound from Erdős's construction.

Limitations and open questions

Several gaps remain open. First, Conjecture ($\sigma(\mathbb{B}^d(0,1))$0 for all symmetric convex $\sigma(\mathbb{B}^d(0,1))$1) is unproved even for the Euclidean ball, and the gap between the lower bound $\sigma(\mathbb{B}^d(0,1))$2 and upper bound $\sigma(\mathbb{B}^d(0,1))$3 for the ball is substantial. Second, Conjecture on $\sigma(\mathbb{B}^d(0,1))$4, if established via Hamming digraphs, would improve the error term in Theorem from $\sigma(\mathbb{B}^d(0,1))$5 to exponentially small; the authors believe a spectral argument for the undirected version is within reach but do not carry it out. Third, the paper poses a stronger question—whether $\sigma(\mathbb{B}^d(0,1))$6 exactly for every centrally symmetric convex $\sigma(\mathbb{B}^d(0,1))$7—and proves a lemma showing that an affirmative answer implies Conjecture, since $\sigma(\mathbb{B}^d(0,1))$8 via a symmetrization argument. Finally, the paper notes that the discrete versions of these problems are strictly harder than the continuous ones treated here.

Conclusion

The paper determines $\sigma(\mathbb{B}^d(0,1))$9 exactly, resolves Bukh's asymptotic question for $\mathbb{R}^d$00-free subsets of spheres with explicit $\mathbb{R}^d$01 error, and establishes a framework connecting these extremal problems to triangle-free subgraphs of unit distance digraphs. The two proof techniques—Brunn-Minkowski discretization and spherical harmonic analysis with ultraspherical polynomial bounds—are each elementary but carefully deployed, and the reduction to $\mathbb{R}^d$02 suggests a concrete program for sharpening the spherical bounds. The central open problem, whether $\mathbb{R}^d$03 is the correct bound for every symmetric convex region, remains unresolved even in the case of the ball.