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 $\mathbf{x},\mathbf{y},\mathbf{z}\in A$ with $\mathbf{x}+\mathbf{y}+\mathbf{z}=0$. 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 $d\to \infty$ 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 $5$.

12 pages