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Source. Lemma 1, p. 2, 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, $\varphi=(1+\sqrt5)/2$.

Lemma 1 (p. 2). Suppose a positive integer $m\notin P(S_t(\alpha))$ and a non-negative integer $r$ satisfy

$$s_1+\cdots+s_r<m<s_{r+2},$$

and $s_n+s_{n+1}\le s_{n+2}$ for every $n>r$. Then for every $k\ge1$,

$$m+s_{r+3}+s_{r+5}+\cdots+s_{r+2k+1}\notin P(S_t(\alpha)).$$

The paper records that Graham (1964) attributes the lemma to Folkman, and that it has found no reference for it (p. 2).

Proof pointer

Induction on $k$ (p. 2): the shifted integer stays strictly between the sum of all earlier terms and the term two places on, so any representation of it must use the newest odd-offset term, and removing that term would represent the previous integer.

Dependencies

None beyond the definitions.

Bears on

• Problem 349: the tool behind Corollary 1, through which the paper proves non-completeness for $\alpha\ge\varphi$; on its own the lemma gives no completeness verdict for any pair.