Rényi stability of $B_h$ sets: a two-order phase diagram and sharp deletion principles

Abstract

A set $B$ in an abelian group is a $B_h$ set if every $h$-term sum has a unique representation up to permutation; for $h=2$ these are the Sidon sets. We study a weighted removal problem for this collision-free property: if the $h$-fold sum map has small Rényi entropy loss, how much probability mass must be deleted to leave a $B_h$ support? Two Rényi orders naturally arise: a collision order $α$, measuring the entropy loss, and a budget order $β$, controlling how spread out the weighting may be. Existing one-order formulations tie the two together on the diagonal $β=α$. We determine the resulting stability problem on the full $(α,β)$-plane. Stability holds exactly when $β\le1$ and $α\geβ$. Inside this region the optimal deletion rate is polynomial for $β<1$ and logarithmic on the boundary $β=1$, where the leading constant is exact; outside it, stability fails through two distinct mechanisms: a supercritical budget and dilution by light atoms. In each case the limiting defect is computed exactly. The upper bounds follow from a sharp list-coarsening inequality with optimal constant, which also yields an entropy-free removal theorem, a finite combinatorial consequence for moments of the representation function, and extensions to $B_h[g]$ sets. Matching constructions show that the phase boundaries and rates are sharp.