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Source. Proposition 5, 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 5 (p. 5). If $\tfrac32\le\alpha<\varphi$, then $S_t(\alpha)$ is entirely complete if and only if $t<\frac3{\alpha^2}$. In particular, if $t\le\frac{9-3\sqrt5}2$, then $S_t(\alpha)$ is entirely complete for all these values of $\alpha$.

The statement concerns entire completeness only: for $t\ge3/\alpha^2$ it says nothing about completeness.

Proof pointer

As in Proposition 4: with $t\ge1$, $s_1=1$ and $s_2=2$, and Lemma 4 with the paper's Lemma 3 gives entire completeness; for $t\ge3/\alpha^2$, $s_2\ge3$, so $2$ is not a subset sum (p. 5).

Dependencies

Lemma 4, Lemma 3 of the paper, and Graham (1964) for $t<1$.

Bears on

• Problem 349: entire completeness implies completeness, so every pair with $\tfrac32\le\alpha<\varphi$ and $t<3/\alpha^2$ gives a complete sequence; the pairs with $t\ge3/\alpha^2$ are left open by this result.