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Source. The two unnumbered statements of the introduction, p. 1, 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 assembly from Propositions 1--6 was checked against their statements. 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, $\varphi=(1+\sqrt5)/2$.

The introduction announces two results, which Section 3 proves through Propositions 1--6.

Entire completeness up to $5^{1/3}$ (p. 1). If $1<\alpha\le5^{1/3}$, then $S_t(\alpha)$ is entirely complete if and only if

$$t<\min\left(\frac2\alpha,\frac3{\alpha^2},\frac5{\alpha^3}\right).$$

Non-completeness from $\varphi$ (p. 1). If $\alpha\ge\varphi$ and

$$t\ge\max\left(\min\left(\frac3{\alpha^2},\frac5{\alpha^3}\right),1\right),$$

then $S_t(\alpha)$ is not complete. The paper states that, combined with Graham's results (R. L. Graham, On a conjecture of Erdős in additive number theory, Acta Arith. 10 (1964), 63--70), this finishes the case $\alpha\ge\varphi$.

Proof pointer

The first statement is the union of Proposition 4 on $[\varphi,5^{1/3})$, Proposition 5 on $[3/2,\varphi)$ and Proposition 6 on $(1,3/2)$, with the endpoint $\alpha=5^{1/3}$ (where $5/\alpha^3=1$) covered by Proposition 3 for $t\ge1$ and by Graham's results for $t<1$. The second is Proposition 1 for $\alpha>2$, Proposition 2 at $\alpha=2$, Proposition 3 on $[5^{1/3},2)$ and the non-completeness half of Proposition 4 on $[\varphi,5^{1/3})$; the non-completeness arguments go through Corollary 1.

Dependencies

Propositions 1--6 of the paper, and Graham's 1964 results for $t<1$.

Bears on

• Problem 349: for every $\alpha\ge\varphi$ the paper, with Graham's results for $t<1$, determines which $t$ give a complete sequence, as it states on p. 1; for $1<\alpha<\varphi$ it determines entire completeness only, and the problem's question stays open there for $t\ge\min(2/\alpha,3/\alpha^2)$ outside the regions of Proposition 7, Proposition 8 and Proposition 9.