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Source. Proposition 2, 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 2 (p. 4). If $\alpha=2$, then $S_t(\alpha)$ is (entirely) complete if and only if $t=1/2^k$ for some $k\ge1$.

The range $k\ge1$ belongs to the paper's indexing from $n=1$; with the sequence indexed from $n=0$ the condition reads $t=1/2^k$ with $k\ge0$.

Proof pointer

For $t=1/2^k$ the nonzero terms are the powers of two. For $t\ge1$, Corollary 1 with $r=0$, $m=1$ applies. For $t<1$ not of that form, the paper takes the first index $n$ with $s_n=1$, writes $t2^n=1+\epsilon$ with $\epsilon\in(0,1)$, and finds a later index where the terms stop being powers of two, to which Corollary 1 applies (p. 4).

Dependencies

Corollary 1.

Bears on

• Problem 349: decides every pair with $\alpha=2$.