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Amazon Web Services（亚马逊云服务）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文在 $(\alpha,\beta)$ 平面上完整刻画了 $B_h$ 集在 Rényi 熵损失下的加权删除稳定性，给出精确的相图、最优删除率及尖锐常数，并推广到 $B_h[g]$ 集。
AI 中文摘要
阿贝尔群中的集合 $B$ 若满足每个 $h$ 项和（在置换意义下）具有唯一表示，则称为 $B_h$ 集；当 $h=2$ 时即为 Sidon 集。我们研究该无碰撞性质的加权删除问题：若 $h$ 重和映射具有较小的 Rényi 熵损失，则必须删除多少概率质量才能留下一个 $B_h$ 支撑集？自然出现两个 Rényi 阶：碰撞阶 $\alpha$（度量熵损失）和预算阶 $\beta$（控制加权的分散程度）。现有的一阶表述在对角线 $\beta=\alpha$ 上将两者联系起来。我们在整个 $(\alpha,\beta)$ 平面上确定由此产生的稳定性问题。稳定性恰好当 $\beta\le1$ 且 $\alpha\ge\beta$ 时成立。在该区域内，最优删除率在 $\beta<1$ 时为多项式阶，在边界 $\beta=1$ 时为对数阶，且前导常数是精确的；在该区域之外，稳定性通过两种不同机制失效：超临界预算和轻原子的稀释。在每种情况下，极限缺陷都被精确计算。上界来自一个具有最优常数的尖锐列表粗化不等式，该不等式还导出一个无熵删除定理、表示函数矩的有限组合推论，以及向 $B_h[g]$ 集的推广。匹配的构造表明相边界和速率是尖锐的。
英文摘要
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 so that the remaining support is a $B_h$ set? Two Rényi orders arise: $α$ is the order at which the coarsening loss is measured, whereas $β$ is the order of the entropy constraint. The diagonal specialization $β=α$ ties the two roles together. We determine the resulting stability problem on the positive $(α,β)$-quadrant. 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 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.
Comments29 pages, 1 figure
