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

A measurable set $A\subseteq {\mathbb{S}}^{d-1}$ is called zero-sum-free if there are no $\mathit{x},\mathit{y},\mathit{z}\in A$ with $\mathit{x}+\mathit{y}+\mathit{z}=\mathbf{0}$. Bukh asked whether every zero-sum-free measurable subset of ${\mathbb{S}}^{d-1}$, for $d\ge 3$, has normalized surface measure at most $\frac{1}{2}$. He also pointed out that even the asymptotic behavior as $d\to \mathrm{\infty }$ was unknown. We answer Bukh’s asymptotic question by proving that every such set has normalized surface measure at most $\frac{\lfloor (d+1{)}^2\mathrm{/}2\rfloor }{d(d+1)}=\frac{1}{2}+O\left(\frac{1}{d}\right)\mathrm{.}$ 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 $5$.