On the maximal measure of a spherical set avoiding solutions to x + y + z = 0

Abstract: We prove that the maximal normalized surface measure of a spherical set in d dimensions avoiding solutions to x + y + z = 0 approaches 1/2 as d goes to infinity. This gives a partial answer to a question of Bukh, who conjectured 1/2 to be the optimal bound for all d >= 3.

• The paper proves that for sets on S^(d-1) avoiding x+y+z=0, the maximal normalized measure is bounded by d/(2(d-1)), converging to 1/2 as d increases.
• It employs a novel combinatorial approach by transforming the geometric problem into a directed graph analysis using random rotations and a generalization of Mantel's theorem.
• The study bridges geometric measure theory with extremal combinatorics, suggesting applications in coding theory and geometric Ramsey theory and outlining future research directions.

Maximal Surface Measure of Spherical Sets Avoiding Solutions to $x + y + z = 0$

Problem Statement and Context

The paper investigates the maximal possible normalized surface measure, $\sigma(A)$, of a measurable subset $A \subset S^{d-1}$ in $\mathbb{R}^d$ that avoids triples of points $x, y, z \in A$ satisfying $x + y + z = 0$. This question arises in the intersection of combinatorial geometry and extremal graph theory, notably as a response to a problem posed by Bukh, who conjectured that the optimal bound for $\sigma(A)$ is $1/2$ for all $d \geq 3$.

Main Theorem and Technical Approach

The principal result states that

$$\sigma(A) \leq \frac{\left\lfloor d^2 / 2 \right\rfloor}{d(d-1)} \leq \frac{d}{2(d-1)},$$

and that the bound is asymptotically sharp: as $\sigma(A)$0, the upper bound approaches $\sigma(A)$1.

The proof leverages a combinatorial translation to directed graphs: for each $\sigma(A)$2, construct a set $\sigma(A)$3 corresponding to directed edges between basis vectors, and interpret the inclusion of $\sigma(A)$4 under random rotations as a random digraph $\sigma(A)$5 with vertex set $\sigma(A)$6. The absence of solutions $\sigma(A)$7 in $\sigma(A)$8 ensures $\sigma(A)$9 is free of directed triangles.

Applying a generalization of Mantel's theorem for digraphs (Brown and Harary, 1970), the maximal number of edges in any directed triangle-free digraph on $A \subset S^{d-1}$0 vertices is $A \subset S^{d-1}$1. Averaging over the rotations, the expected number of edges induced by $A \subset S^{d-1}$2 is $A \subset S^{d-1}$3, yielding the desired bound.

Crucially, this approach converts a geometric extremal question into a combinatorial extremal problem via representation theory and probabilistic averaging over group actions, providing a rigorous framework for bounding $A \subset S^{d-1}$4.

Numerical Results and Asymptotics

For finite $A \subset S^{d-1}$5, the explicit bound is

$A \subset S^{d-1}$6

which is tight for $A \subset S^{d-1}$7, as

$A \subset S^{d-1}$8

The result confirms that $A \subset S^{d-1}$9 is achievable in all dimensions, establishing the asymptotic correctness of the conjectured bound.

Implications and Future Directions

This result links discrete combinatorial extremal theory with continuous geometric measure theory, strengthening the understanding of forbidden configurations on high-dimensional spheres. The construction demonstrates the utility of group actions (rotations) in transferring combinatorial bounds to geometric settings.

Potential avenues for future research include:

• Tightening the bound for finite $\mathbb{R}^d$0 or determining whether the bound $\mathbb{R}^d$1 is always attainable for every $\mathbb{R}^d$2.
• Extending the forbidden configurations from $\mathbb{R}^d$3 to more general additive relations or higher-order analogs.
• Application to coding theory and geometric Ramsey theory, where avoiding certain additive or combinatorial patterns is central.

The methods may further inspire results in high-dimensional extremal combinatorics via geometric or probabilistic representations.

Conclusion

The paper establishes that for measurable subsets of $\mathbb{R}^d$4 avoiding solutions to $\mathbb{R}^d$5, the maximal normalized surface measure asymptotically converges to $\mathbb{R}^d$6 as $\mathbb{R}^d$7 increases, with explicit bounds valid for finite dimensions. The approach exploits extremal digraph theory and rotational symmetry, contributing foundational advances to the interplay between combinatorial and geometric extremal problems, and opening new prospects for research in additive avoidance on spheres (2607.05083).