Scott Duke Kominers
5 min
Abstract
Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0<c\le 1$, the bound $n>6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.
Sam: That's the angle. For dimension up to about 1.17 times ten to the five times c squared, the new bound is smaller. It replaces sequential convexification—one summand per tree generation, losing thickness each time—with simultaneous processing for all sets at once, via a radius form of the Shapley-Folkman theorem from economics and geometry. That turns multiplicative losses into one additive error at each scale, shared across all summands.
Alex: Oh—so instead of errors compounding like interest over steps, it's a flat fee that the group pays together. But before we get into that theorem, walk me through thickness again—why multiscale?
Sam: Thickness checks locally at every point x and radius r up to the set's diameter: exists y such that a ball of size proportional to c r sits inside the convex hull of the set near x within r. Multiscale because fractals look similar at all sizes, like a coastline that stays jagged no matter how you zoom in, so you need this at every scale to capture structure. Compactness lets you discretize to finite points without losing much.
Alex: Got it—like ensuring no matter how you zoom in, there's always a solid chunk nearby. So the old proof used tree decompositions but sequentially, racking up slack. How does the new one handle the tree differently?
Sam: The proof builds recursive trees for each set, starting at a coarse scale, then refining to finer scales. At each level, finite points whose convex hull holds a decent-sized ball. But the key shift is applying an absorption step to all sets at once, using Shapley-Folkman to bridge the sum of convex hulls to the actual sum of points within a controlled error. With enough sets, that error gets absorbed.
Alex: Wait—so the monotonicity across levels comes from that one-step absorption across all summands. And the limit traps a ball inside the actual sum.
Sam: Precisely. Parameters are tuned so alpha is near c and the refinement factor minimizes the threshold. No packing arguments needed—just compactness for finites and stability tools.
Alex: So the practical challenge was sequential bottlenecks losing thickness per step, cubic overall; now parallel pipelines with fixed leakage per level, quadratic suffices. That seems like a clear improvement for high-d or small-c sums in Diophantine stuff or Palis.
Sam: Yes, and while sqrt(d) comes from uniform Shapley-Folkman on tree clouds, their structure might sharpen it further. The paper notes it's not dimension-free yet and assumes positive diameter. But for fixed d, quadratic dominates cubic for small c, opening analysis where old bounds failed.
Alex: Makes sense—this transforms how we bound these sums. Thanks for laying it out so clearly, Sam. Thanks for listening to ResearchPod.