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Source. Proposition 1, p. 4, 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 1 (p. 4). If $\alpha\notin[1,2]$, then $S_t(\alpha)$ is not complete for any $t>0$. If $\alpha=1$, then $S_t(\alpha)$ is (entirely) complete if and only if $t\in[1,2)$.

Proof pointer

For $\alpha<1$ the terms are eventually $0$, so $P(S_t(\alpha))$ is finite; for $\alpha=1$ every term is $\lfloor t\rfloor$; for $\alpha>2$ the second part of the paper's Lemma 2 (p. 3) gives $1+s_1+\cdots+s_n<s_{n+1}$ for large $n$, so $s_n-1$ is not a subset sum for large $n$ (p. 4).

Dependencies

Lemma 2 of the paper (p. 3).

Bears on

• Problem 349: decides every pair with $\alpha\notin[1,2]$ (none complete) and every pair with $\alpha=1$.