---

Source. Proposition 9, p. 10, 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 proof was read for structure only. 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 9 (p. 10). The sequence $S_t(\alpha)$ is complete for all $\alpha$ with

$$1<\alpha\le1+\frac1{\lceil t\rceil+2\lceil\sqrt t\rceil}.$$

The introduction (p. 2) describes the same result with a strict inequality on $\alpha$; the proposition as printed has $\le$.

Proof pointer

With $v=\lceil t\rceil$ and $w=\lceil\sqrt t\rceil$, the paper's Lemma 7 puts every integer of $[v,v+2w]$ in $S_t(\alpha)$ and its Lemma 6 bounds the next term; sums of $w$ and of $w+1$ of these terms cover a run of consecutive integers long enough for Lemma 5 (pp. 10--11). The proof assumes $t\ge1$, which the paper says it is free to do (p. 11).

Dependencies

Lemma 5, Lemmas 6 and 7 of the paper (pp. 6--7).

Bears on

• Problem 349: every pair in this unbounded region gives a complete sequence.