Year: 2026
Type: preprint
Abstract: Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. As an application, we show that a uniformly at random chosen subset $A \subseteq \{1, \ldots, n\}$ of size $k$ such that $|A + A| \le λk$, 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.
Source: arXiv (Cornell University)
Authors: Nenadov, Rajko, Verlinde, Lander
Institution: University of Canterbury
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162767400