The sharp asymptotic density of zero-sum-free spherical sets

Abstract: A measurable set $A\subseteq \mathbb S<sup>{d-1}$ is called zero-sum-free if there are no $\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in A$ with $$\boldsymbol{x}+\boldsymbol{y}+\boldsymbol{z}=\boldsymbol{0}$$. Bukh asked whether every zero-sum-free measurable subset of $\mathbb S<sup>{d-1}$, for $d\ge3$, has normalized surface measure at most $\frac{1}{2}$. He also pointed out that even the asymptotic behavior as $d\to\infty$ was unknown. We answer Bukh's asymptotic question by proving that every such set has normalized surface measure at most $$\frac{\lfloor (d+1)<sup>2/2\rfloor}{d(d+1)}=\frac{1}{2}+O\left(\frac{1}{d}\right).$$ Since the lower bound $\frac{1}{2}$ comes from open hemispheres, this determines the asymptotic extremal density. By monotonicity, upper bounds in low-dimensional cases are especially important. We use a stability argument to improve the bound from $\frac{3}{5}$ to $\frac{71}{120}$ in dimensions $4$ and $5$.

• The paper establishes that any zero-sum-free measurable subset on the sphere has density at most 1/2 + O(1/d), matching the lower bound from hemispheres.
• It translates the geometric extremal problem into a combinatorial one using a reduction to triangle-free digraphs and probabilistic averaging over rotations.
• It improves explicit bounds in low dimensions (d=4,5) via a detailed stability analysis of symmetric configurations like the H4 root system.

The Sharp Asymptotic Density of Zero-Sum-Free Spherical Sets

Problem Setting and Motivations

This paper addresses Bukh's question on the maximal density of "zero-sum-free" measurable subsets on the unit sphere $\mathbb S^{d-1} \subseteq \mathbb R^d$: a set $A \subseteq \mathbb S^{d-1}$ is zero-sum-free if it does not contain three points $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$ with $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$. The supremum of the surface measure $\sigma_{d-1}(A)$ over all such $A$ is denoted $m_d$.

Key motivations derive from extremal combinatorics and geometric Ramsey theory. The problem is a continuous analogue of the cap set problem in $\mathbb F_3^n$, which has had deep connections with the polynomial method and additive combinatorics. The sphere lacks translation symmetry, distinguishing spherical configurations from their finite field counterparts and implicating rich rotational symmetries and geometric considerations.

Main Results

The paper achieves two primary results regarding the upper bounds on the density $m_d$ of zero-sum-free spherical sets:

• Sharp Asymptotic Upper Bound: For all $d \geq 2$, all measurable zero-sum-free $A \subseteq \mathbb S^{d-1}$0 satisfy

$A \subseteq \mathbb S^{d-1}$1

This shows $A \subseteq \mathbb S^{d-1}$2, matching the lower bound given by open hemispheres. This completely resolves the asymptotic question posed by Bukh.

• Improved Bounds in Low Dimensions: For $A \subseteq \mathbb S^{d-1}$3, an explicit stability argument strengthens this bound from $A \subseteq \mathbb S^{d-1}$4 to $A \subseteq \mathbb S^{d-1}$5, demonstrating that $A \subseteq \mathbb S^{d-1}$6.

These findings settle the extremal density problem for large dimensions and provide substantial progress for small $A \subseteq \mathbb S^{d-1}$7.

Technical Approach

Reduction to Extremal Digraphs

A core insight is the encoding of zero-sum-free sets via directed triangle-free digraphs. The analysis leverages a spherical configuration (the normalized $A \subseteq \mathbb S^{d-1}$8 root system), relating points on the sphere to directed edges in a complete digraph $A \subseteq \mathbb S^{d-1}$9 with $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$0. Zero-sum triples on the sphere correspond exactly to directed triangles in $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$1.

Applying a probabilistic method—averaging over rotations of the spherical configuration—the authors translate the geometric extremal problem into an extremal problem for digraphs. They use the Brown-Harary directed version of Mantel's theorem: A triangle-free digraph on $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$2 vertices has at most $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$3 edges. This combinatorial result is then mapped to the spherical setting to yield the sharp bound for general $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$4.

Stability Arguments for $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$5

In the low-dimensional cases, the authors employ stability and finite averaging techniques using highly symmetric finite point configurations (e.g., the $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$6 root system for $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$7), enabling an intricate combinatorial analysis. The argument depends on exhaustive enumeration of zero-sum-free subsets within these specialized configurations and sophisticated use of incidence matrices and eigenvalue analysis of the associated graphs.

The key lemma shows that in the $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$8 configuration (120 points in $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$9), a zero-sum-free subset can have at most $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$0 points; this, combined with averaging over rotations, leads directly to the improved upper bound of $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$1.

Strong Claims and Notable Features

• Sharp Asymptotics: The claim $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$2 is tight, as hemispheres yield the lower bound $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$3.
• Finite Configuration Methods: The structure and analysis of the $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$4 root system for $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$5 and use of combinatorial properties of the associated graphs produce the best known bounds for small $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$6.
• Monotonicity: The proof includes a monotonicity lemma, implying that improved bounds in small dimensions propagate to larger ones.

Implications and Prospects

Theoretical Implications

The solution to the spherical zero-sum-free extremal problem not only resolves a question parallel to classical sum-free and cap set phenomena in additive combinatorics, but also fortifies the link between geometric configurations and extremal graph theory. The mapping from spherical geometry to extremal digraphs suggests broader applicability to geometric problems with forbidden configurations, including those involving codes, designs, and independence densities.

The approach also showcases the power of symmetries and probabilistic averaging in continuous combinatorial geometry, opening prospects for similar methods in problems where group actions and extremal combinatorics intersect.

Practical Implications and Future Directions

While the main results are foundational, the techniques used (especially the reduction to rotational averaging and combinatorial finiteness arguments) may influence algorithms for point set design, error-correcting codes, and spherical packing in higher dimensions.

Future developments may include:

• Finer Bounds in Small Dimensions: Given the tight asymptotics, it remains to close the remaining gap for $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$7 and further sharpen estimates for $$\boldsymbol{x} + \boldsymbol{y} + \boldsymbol{z} = \boldsymbol{0}$$8.
• Generalizations to Forbidden Configurations: There is potential for extension to sets avoiding more complicated configurations, angles, or relations on the sphere or other symmetric spaces.
• Transfer to Discrete and Finite Geometries: Analogous techniques could yield new results in finite field geometry and combinatorial design theory.

Conclusion

This paper establishes the sharp asymptotic density for zero-sum-free measurable sets on spheres, resolving a longstanding extremal problem and uniting combinatorial, geometric, and algebraic techniques. The results delineate the boundary between geometric extremality and additive combinatorial rigidity, with structural methods that are likely to be influential in related geometric and combinatorial extremal questions (2607.05099).