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Claim. The answer to Problem 337 is no: there is an additive basis $A\subseteq\mathbb{N}$ of finite order with $\lvert A\cap\{1,\ldots,N\}\rvert=o(N)$ for which

$$\liminf_{N\to\infty}\frac{\lvert (A+A)\cap\{1,\ldots,N\}\rvert}{\lvert A\cap\{1,\ldots,N\}\rvert}<\infty.$$

The result is S. Turjányi, A note on basis sequences, in Topics in classical number theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. János Bolyai 34 (1984), 1571--1576, which constructs such a basis of order $k$ for every $k\ge4$. The note is not held here; its content is known through the introduction of Ruzsa and Turjányi's 1985 paper, which records that Turjányi disproved the Erdős--Graham conjecture by this construction (source card), and through the site's commentary.

Acceptance. Reviewed: the site's curator, T. F. Bloom, credits the negative answer to this note and labels the problem disproved at erdosproblems.com (label), which is the site's acceptance. The note appeared in a colloquium proceedings volume rather than a journal, so no refereed evidence is listed; the acceptance recorded here is the site curator's alone. Ruzsa and Turjányi's paper in Publ. Math. Debrecen 32 (1985), co-written by the claimant, records the note as the disproof and proves a generalization whose Theorem 1 covers the note's statement, but as the claimant's own paper it is context rather than independent acceptance; that paper's own counterexamples of every order $h\ge3$ are recorded at Ruzsa and Turjányi 1985. The proof is not compiled or reviewed here.

Context. The problem is Problem 2 at the close of Erdős and Graham's paper On bases with an exact order, Acta Arith. 37 (1980), 201--207 (source card), the site's [ErGr80b], which asks, among questions the authors could not settle, whether the limit is infinite; Ruzsa and Turjányi cite it as Erdős and Graham's conjecture, and the site also cites Erdős and Graham's 1980 monograph [ErGr80]. The page's date is the proceedings volume's year of publication, 1984, with the day set to the year's first since the volume carries no finer date; the result was presented at the 1981 colloquium.

Depends on. No page of this wiki.