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Claim. For $t,a>0$ let $S_t(a)$ be the sequence with $n$th term $\lfloor ta^n\rfloor$, each index used at most once (the paper's $P(A)$ is the set of sums $\sum\varepsilon_ka_k$ with $\varepsilon_k\in\{0,1\}$, almost all zero). Graham determines the set $T$ of pairs $(t,a)$ in the unit square $0<t<1$, $1<a<2$ for which $S_t(a)$ is complete and shows that $T$ has area approximately $0.85$, so Erdős's conjecture that $S_t(a)$ is complete for every $t>0$ and $1<a<2$ is false. Theorem 2: for $0<t<1$ and $1<a\le5^{1/3}$ the sequence is entirely complete (every positive integer is a sum). Theorem 3: for $0<t<1$ and $1<a<2$ the sequence is complete if and only if it is entirely complete. The site's remarks add the paper's consequence that for every $k$ there is $t_k\in(0,1)$ for which the set of $a$ with $S_{t_k}(a)$ complete has at least $k$ connected components. The paper is Graham, R. L., On a conjecture of Erdős in additive number theory, Acta Arith. 10 (1964/65), 63--70, recorded on the card graham_nd_conjecture_erdos_additive_number_theory.

Covers. The pairs with $0<t<1$ and $1<\alpha<2$ of the corrected Statement of Problem 349, whose sums (each term used at most once, equal values at different indices counted separately) and index from $n=1$ are the paper's: on that square the result determines exactly which pairs give a complete sequence, which is what the problem asks, so the claim's value is answered. It says nothing about $t\ge1$ or about $\alpha\ge2$ or $\alpha\le1$; those regions are the subject of van Doorn's claim page, which builds on this one.

Acceptance. The refereed evidence is the journal publication cited above, in Acta Arithmetica. The site's curator cites the paper in the remarks of a problem the site labels OPEN, which credits the partial result without settling the problem, so no reviewed evidence is listed. The record gives the volume years and no finer date, so the page is dated to the first day of 1964.

Depends on. Nothing in this wiki.