Rényi stability of Bh sets: a two-order phase diagram and sharp deletion principles
Abstract
A set B in an abelian group is a Bh 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 Bh 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≤1 and α≥β. 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 Bh[g] sets. Matching constructions show that the phase boundaries and rates are sharp.
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BibTeX
@misc{woo2026rényi,
title = {{Rényi stability of $B_h$ sets: a two-order phase diagram and sharp deletion principles}},
author = {Jae Oh Woo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23922},
eprint = {2609.23922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23922},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rényi stability of B_h sets: a two-order phase diagram and sharp deletion principles
AU - Jae Oh Woo
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.23922
UR - https://arxiv.org/abs/2609.23922
AB - 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≤1 and α≥β. 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.
ER -