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Source. Proposition 8, p. 9, of Wouter van Doorn, Completeness of exponentially increasing sequences, arXiv:2602.23394v1 (25 February 2026), the version named on the source card. A preprint.

Read depth. Claims checked: the statement was read clause by clause on the page images of the print; the description of the search was read, and the search data were not run. Nothing here is independently reviewed.

Statement

Setting (p. 1). For positive reals $t$ and $\alpha$, $S_t(\alpha)=(s_1,s_2,\ldots)$ with $s_n=\lfloor t\alpha^n\rfloor$, indexed from $n=1$. For a sequence or multiset $S$ of positive integers, $P(S)$ is the set of integers that are sums of distinct elements of $S$; $S$ is complete when $\mathbb N\setminus P(S)$ is finite and entirely complete when $P(S)=\mathbb N$. Throughout, $\varphi=(1+\sqrt5)/2$.

Proposition 8 (p. 9). $S_t(\alpha)$ is complete

• for all $t\le3$ if $1.3<\alpha\le1.4$;
• for all $t\le5$ if $1.2<\alpha\le1.3$;
• for all $t\le10$ if $1.1<\alpha\le1.2$;
• for all $t\le50$ if $1<\alpha\le1.1$.

Proof pointer

Computer-assisted (pp. 9--10). The region is covered by rectangles; for each rectangle the terms common to the sequences at its bottom-left and top-right corners, together with the range of one further term, are shown to meet the hypothesis of Lemma 5 for every pair in the rectangle, and a rectangle that fails is split into four. The partitions are published in the author's repository (https://github.com/Woett/Complete-sequences-data). For $\alpha\le1.2$ the search uses $t\ge2$ by Proposition 7, and for $\alpha\le1.1$ it uses $\alpha\ge1.015$ by Proposition 9 (p. 10).

Dependencies

Lemma 5, Proposition 7, Proposition 9, and the published search data.

Bears on

• Problem 349: every pair in the four stated boxes gives a complete sequence, on the strength of the computer search; the search data were not checked here.