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

Abstract

A measurable set $A\subseteq \mathbb S^{d-1}$ 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 $\mathbb S^{d-1}$, 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 $\frac{\lfloor (d+1)^2/2\rfloor}{d(d+1)}=\frac{1}{2}+O\left(\frac{1}{d}\right).$ 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 $5$.