Improved bounds for the double cap conjecture

Abstract

In 1974, Witsenhausen asked for the maximum possible density $α_n$ of a measurable subset $A$ of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known lower bound is $1 - 1/\sqrt{2} = 0.29289\dots$, obtained from the natural "double cap" construction of two opposite spherical caps, which is conjectured to be optimal for all $n$ by Gil Kalai. In this paper, we use a novel approach to establish an upper bound of $α_3\le 0.2953$, improving the previous best known bound $0.2977$ due to Bekker et al. (2025). Our approach combines harmonic-analytic arguments with the geometric fractional chromatic number of finite graphs, recently introduced by Ambrus et al. (2024). In this framework, any finite subset of the sphere yields an upper bound for $α_n$, and we obtain our bound by identifying an appropriate 33-element point set through a large-scale computer search. The same method can also be used in higher dimensions to yield potential improvements of the best known bounds.