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Source. Proposition 3, 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 3 (p. 5). If $5^{1/3}\le\alpha<2$, there is no value of $t\ge1$ for which $S_t(\alpha)$ is complete.

Proof pointer

Three cases on $t$ (p. 5): $t\ge2/\alpha$ gives $s_1\ge2$ and Corollary 1 with $m=1$, $r=0$; otherwise $t\ge3/\alpha^2$ gives $s_2\ge3$ and Corollary 1 with $m=2$, $r=1$; otherwise $t\ge1\ge5/\alpha^3$ gives $s_3\ge5$ and Corollary 1 with $m=4$, $r=2$.

Dependencies

Corollary 1.

Bears on

• Problem 349: decides every pair with $5^{1/3}\le\alpha<2$ and $t\ge1$ (none complete); the pairs with $t<1$ there are Graham's (1964).