Problem 337

Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$ . Is it true that

\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?

Source: erdosproblems.com/337

Number theory
Additive combinatorics
Additive bases

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SolvedDisproved

An accepted solution exists. The statement is false.

Disproved, the site's label (DISPROVED (LEAN),): Turjányi [Tu84] gave a counterexample basis of every order $k\ge4$ , Ruzsa and Turjányi [RT85] gave one of every order $h\ge3$ with the $(h-1)$ -fold sumset in place of $A+A$ , and their order-3 construction has a Lean 4 proof, the label's Lean qualifier. The accepted claims are Turjányi 1984 and Ruzsa and Turjányi 1985.

Problem 337

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