R\'enyi 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 R\'enyi entropy loss, how much probability mass must be deleted so that the remaining support is a $B_h$ set? Two R\'enyi orders arise: $\alpha$ is the order at which the coarsening loss is measured, whereas $\beta$ is the order of the entropy constraint. The diagonal specialization $\beta=\alpha$ ties the two roles together. We determine the resulting stability problem on the positive $(\alpha,\beta)$-quadrant. Stability holds exactly when $\beta\le1$ and $\alpha\ge\beta$. Inside this region the optimal deletion rate is polynomial for $\beta<1$ and logarithmic on the boundary $\beta=1$, where the leading constant is exact; outside it, stability fails through two distinct mechanisms: a supercritical budget and dilution by light atoms. In both unstable regimes the limiting defect is computed exactly. The upper bounds follow from a coarsening inequality with best possible 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.