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Statement

Definitions (printed p. 24): for a sequence $S=(s_1,s_2,\ldots)$ of positive integers, $P(S)$ is the set of all finite sums $\sum_k\varepsilon_ks_k$ with $\varepsilon_k\in\{0,1\}$, and $S$ is complete "if all sufficiently large integers belong to $P(S)$". A sequence may repeat a value, and repeated values count separately in $P(S)$. Square brackets denote the integer part.

Question 12 (printed p. 36, quoted). "Let $\alpha$ and $\beta$ be positive reals with $\alpha/\beta$ irrational. Let $S$ denote the sequence $([\alpha],[\beta],[2\alpha],[2\beta],\ldots,[2^n\alpha],[2^n\beta],\ldots)$. Is $S$ complete? What if $2$ is replaced by some $\gamma$, $1<\gamma<2$?"

This is the statement of Problem 354 up to notation: the site's multiset $\{\lfloor2^s\alpha\rfloor\}\cup\{\lfloor2^t\beta\rfloor\}$ is the paper's sequence $S$, and the site's second question is the paper's last sentence. The paper offers no result on either question. Its nearest context is Question 2 (printed p. 34): for which $(t,\alpha)$ with $t>0$, $1<\alpha<2$ is the sequence $s_n=[t\alpha^n]$ complete, known for $0<t\le1$ from the author's 1964 Acta Arithmetica paper and, quoted, "Even in the range $1<t<2$, it is not known what happens. Conceivably, $S$ is complete for all $1<\alpha<\frac{1+\sqrt5}2$ and $t>0$."

Source. R. L. Graham, On sums of integers taken from a fixed sequence, Proceedings of the Washington State University Conference on Number Theory (1971), 22--40; Question 12 on printed p. 36 = PDF p. 15, the definitions on printed p. 24 = PDF p. 3 and Question 2 on printed p. 34 = PDF p. 13 of the author's publication-page scan, read on the page images (the scan has no text layer). The artifact is identified in the source digest.

Read depth. Claims checked: the question, the definitions and Question 2 were read clause by clause on the page images. A question; the paper proves nothing about it. Nothing here is independently reviewed.

Proof pointer

None. The paper asks the question and stops.

Dependencies

None.

Bears on

• Problem 354: the problem's origin, in the wording the site's statement follows, including the second question with $2$ replaced by $\gamma\in(1,2)$.