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Source. Proposition 7, p. 6, 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 7 (p. 6). If $1<\alpha\le\tfrac54$, then $S_t(\alpha)$ is complete for all $t<\frac4\alpha$.

Proof pointer

Proved on pp. 7--8 by case analysis on which small integers occur in $S_t(\alpha)$, using the paper's Lemma 6 ($s_n<\alpha(s_{n-1}+1)$ for $n\ge2$) and Lemma 7 (if $1<\alpha\le1+1/x$ for some $x>t$, every integer in $[s_1,x]$ is a term), and closing each case with Lemma 5.

Dependencies

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

Bears on

• Problem 349: every pair with $1<\alpha\le5/4$ and $t<4/\alpha$ gives a complete sequence, a region below $\varphi$ beyond the entire-completeness range of Proposition 6.