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Source. Proposition 4, p. 5, 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 4 (p. 5). If $\varphi\le\alpha<5^{1/3}$, then $S_t(\alpha)$ is (entirely) complete if and only if $t<\min\left(\frac3{\alpha^2},\frac5{\alpha^3}\right)$.

Proof pointer

For $t\ge\min(3/\alpha^2,5/\alpha^3)$ the argument of Proposition 3 gives non-completeness. For smaller $t$, Theorem 2 of Graham (1964) allows $t\ge1$; then $s_1=1$, $s_2=2$, $s_3=4$, and Lemma 4 with the paper's Lemma 3 (Graham's Lemma 2, p. 4) gives entire completeness (p. 5).

Dependencies

Corollary 1, Lemma 4, Lemma 3 of the paper, and Theorem 2 of Graham (1964).

Bears on

• Problem 349: decides every pair with $\varphi\le\alpha<5^{1/3}$.