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The typical structure of sets with large sumset
Abstract
Given m ∈ ℕ and a p-random subset A ⊆ ℕ, we asymptotically determine log Pr(|ℕ ∖ (A + A)| ≥ m) for p above the threshold for this property. As an application, we show that a uniformly at random chosen subset A ⊆ {1, …, n} of size k such that |A + A| ≤ λk, for λ= o(k1/4 / log1/2 n), with high probability belongs to an arithmetic progression of length λk / 2 + o(λk). Both proofs are based on a bespoke container argument, exploiting the additive robustness of the considered sets.
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BibTeX
@misc{nenadov2026typical,
title = {{The typical structure of sets with large sumset}},
author = {Rajko Nenadov and Lander Verlinde},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.29680},
eprint = {2605.29680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29680},
}
RIS
TY - UNPB
TI - The typical structure of sets with large sumset
AU - Rajko Nenadov
AU - Lander Verlinde
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.29680
UR - https://arxiv.org/abs/2605.29680
AB - Given m ∈ ℕ and a p-random subset A ⊆ ℕ, we asymptotically determine log Pr(|ℕ ∖ (A + A)| ≥ m) for p above the threshold for this property. As an application, we show that a uniformly at random chosen subset A ⊆ {1, …, n} of size k such that |A + A| ≤ λk, for λ= o(k^1/4 / log^1/2 n), with high probability belongs to an arithmetic progression of length λk / 2 + o(λk). Both proofs are based on a bespoke container argument, exploiting the additive robustness of the considered sets.
ER -