The site's parenthesis asks for sums of distinct integers of the form $\lfloor t\alpha^n\rfloor$, so a value taken at several indices can be used only once. That changes the answer. At $t=1$, $\alpha=1$ every term is $1$: under the site's wording the only sums are $0$ and $1$, so the sequence is not complete, while with each term usable once every positive integer is a sum; the same holds for every $1\le t<2$ at $\alpha=1$. Below $\alpha=2$ coincident values matter as well: at $t=2/3$, $\alpha=1.7$ the terms begin $s_1=s_2=1$, $s_3=3$, and Graham's Theorem 2 makes the sequence entirely complete only when both ones are available, as Wouter van Doorn pointed out in the site's thread on 2025-09-07. The site's wording also names no first index, which changes the answer too: with the index from $n=0$ the pair $(1,2)$ gives $1,2,4,\ldots$ and is complete, and with the index from $n=1$ it gives $2,4,8,\ldots$ and is not.

The change replaces "the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$" by "of the form $\sum_{n\ge1}\varepsilon_n\lfloor t\alpha^n\rfloor$ with $\varepsilon_n=0$ or $1$ and $\sum_{n\ge1}\varepsilon_n<\infty$" and inserts "$n\ge1$" after the sequence; nothing else changes. The evidence is the posers' own text, the passage the site cites. Erdős and Graham [ErGr80] (Old and new problems and results in combinatorial number theory), printed p. 57, ask: "Let $S(t,\alpha)=(s_1,s_2,\ldots)$ with $s_n=[t\alpha^n]$. For what values of $t$ and $\alpha$ is $S(t,\alpha)$ complete?" Their chapter on completeness defines, printed p. 53, $P(A)=\{\sum_{k=1}^\infty\varepsilon_k\alpha_k:\varepsilon_k=0$ or $1,\ \sum_{k=1}^\infty\varepsilon_k<\infty\}$ for a sequence $A=(\alpha_1,\alpha_2,\ldots)$, notes on printed p. 54 that these sums "are restricted by the multiplicity any particular term can have", and calls a sequence $S=(s_1,s_2,\ldots)$ of integers complete "if $P(S)$ contains all sufficiently large integers" (printed p. 54). Graham's paper [Gr64e], §§1--2 (its card), states Erdős's conjecture for $S_t(a)=(s_1,s_2,\ldots)$, $s_n=[ta^n]$, with the same sums. No text of the posers counts values once or starts the index at $n=0$, so the defect is the site's. The monograph's card does not transcribe the printed p. 57 passage. The form follows the posers' statement of this question, not the results that settle parts of it, and the change moves no standing: the problem is open under the site's wording and under the corrected Statement.

Results about the site's wording are credited here and count for nothing. The Lean proofs contributed to formal-conjectures under the account cepadugato in June 2026 (pull requests #4225 and #4233, proofs pinned at 23c629bc and 19e39e33) state the site's wording, with values counted once and the index from $n=0$. Their non-completeness results for $0<\alpha\le1$ and for positive integer pairs other than $(1,2)$ answer only that wording; their completeness results at $\alpha=2$ carry over to the Statement, and their claim page (cepadugato, 2026) records them with that scope.