---

Claim. Let $\alpha,\beta>0$, where $\alpha$ is a dyadic rational, a rational number of the form $m/2^n$, and $\beta$ is not. Then the interleaved sequence $\lfloor\alpha\rfloor,\lfloor\beta\rfloor,\lfloor2\alpha\rfloor,\lfloor2\beta\rfloor,\ldots$ is complete: every sufficiently large integer is $\sum_{s\in S}\lfloor2^s\alpha\rfloor+\sum_{t\in T}\lfloor2^t\beta\rfloor$ for some finite $S,T\subset\mathbb N$. The paper is Hegyvári, N., Some remarks on a problem of Erdős and Graham, Acta Math. Hungar. 53 (1989), 149--154. Neither the paper nor a review of it was read for this page. The theorem is stated as Hegyvári's later paper On complete sequences, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 34 (1991), 7--10 (library card), reports it on p. 7: in the 1989 paper he "settled the Erdős-Graham conjecture when $\alpha$ is a finite diadical fraction and $\beta$ is an infinite diadical fraction", and stated the stronger conjecture that the sequence is complete whenever $\beta/\alpha\ne2^m$ and $\alpha$ is an infinite dyadic fraction. Fan's preprint (Strongly complete sets and a conjecture of Erdős, arXiv:2607.14071v5, p. 4) and the Yu--Chen manuscript of 13 September 2026 (p. 1) give the same account. The 1989 paper also proves that the sequence is not complete when $\alpha\ge2$ and $\beta=2^k\alpha$; that ratio is rational, so the incompleteness result settles no instance of the problem, and it is recorded on the problem page.

Covers. The first question of Problem 354 for $\alpha$ a dyadic rational and $\beta$ irrational, or the reverse; there $\alpha/\beta$ is irrational and the answer is yes. The theorem also covers rational $\beta$ that are not dyadic, which lie outside the problem's irrationality hypothesis. Nothing is covered when neither coefficient is a dyadic rational, nor for the second question, a base $\gamma\in(1,2)$ in place of $2$.

Acceptance. The refereed evidence is the journal publication cited above, in Acta Mathematica Hungarica. The site labels the problem OPEN, so its remarks credit the paper without settling the problem and no reviewed evidence is listed. The publication record dates the issue, volume 53, no. 1--2, to March 1989 and gives no finer date, so the page is dated to the first day of that month.

Depends on. Nothing in this wiki.