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Source. Lemma 4, 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 cited part of Graham's Lemma 4 was not read. 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, $\varphi=(1+\sqrt5)/2$.

Lemma 4 (p. 4). Assume $t\ge1$. Then $s_{n+1}\le2s_n$

• for all $n\ge1$ if $1<\alpha<\tfrac32$;
• for all $n\ge2$ if $\tfrac32\le\alpha<\varphi$;
• for all $n\ge3$ if $\varphi\le\alpha<5^{1/3}$.

Proof pointer

The paper states that the claims follow from the first part of Lemma 4 of Graham (1964) and gives no further proof (p. 4).

Dependencies

Lemma 4 of R. L. Graham, On a conjecture of Erdős in additive number theory, Acta Arith. 10 (1964), 63--70.

Bears on

• Problem 349: the doubling bound that Lemma 5 and Propositions 4--6 use to propagate a run of subset sums; it decides no pair on its own.