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Source. Section 4, Remarks, p. 655, of J. Folkman, On the representation of integers as sums of distinct terms from a fixed sequence, Canad. J. Math. 18 (1966), 643--655, doi:10.4153/CJM-1966-065-2. The edition read is identified on the source card.

Statement

In the paper "increasing" means $a_1\le a_2\le\cdots$ and "strictly increasing" means $a_1<a_2<\cdots$; $A$ is subcomplete when its set $P(A)$ of sums of distinct terms contains an infinite arithmetic progression.

Counterexamples for $\alpha>1$ (p. 655). Let $\alpha>1$. The paper says that it is easy to construct an increasing sequence with $a_n\le n^{\alpha}$ for which

$$\text{(4.1)}\qquad \sup_n\Bigl(a_{n+1}-\sum_{i=1}^{n}a_i\Bigr)=\infty,$$

and such a sequence is not subcomplete; a similar construction gives a strictly increasing sequence satisfying (4.1) and $a_n\le n^{1+\alpha}$. The paper concludes that its theorems, among them Theorem 1.3, are false for $\alpha>1$. The constructions are described, not written out. The paper adds that Cassels (Acta Sci. Math. Szeged 21 (1960), 111--124) constructs counterexamples to Theorem 1.2 and to the strictly increasing case of Theorem 1.3 for $\alpha>1$ that also satisfy $a_{n+1}=a_n+o(a_n^{1/2+\epsilon})$ for an arbitrary preassigned $\epsilon>0$.

Open questions (p. 655). The paper leaves open:

1. Whether every increasing sequence with $a_n\le Mn$ for all $n$ is subcomplete.
2. Whether every strictly increasing sequence with $a_n\le Mn^2$ for $n\ge n_0$, where $M\le1/2$, is subcomplete. The paper notes that $M\le1/2$ is required so that $A$ cannot satisfy (4.1).

The two questions are the boundary case $\alpha=1$ of the growth conditions (1.1) and (1.3), which neither the theorems ($0\le\alpha<1$) nor the counterexamples ($\alpha>1$) cover.

Read depth. Claims checked: the section was read clause by clause on the print. The constructions and Cassels's paper were not read.

Dependencies

None in the corpus. External: Cassels's counterexamples, cited, not proved.

Bears on

• Problem 343: the first open question is the problem's question in Folkman's form, for a nondecreasing sequence with $a_n\le Mn$, the constant $M$ depending on the sequence. The paper gives no answer. Its increasing counterexample with $a_n\le n^{\alpha}$, $\alpha>1$, has at least $\lfloor N^{1/\alpha}\rfloor$ terms up to $N$, so counting functions of order $N^{1-\epsilon}$ do not force subcompleteness.
• Problem 344: the second open question concerns strictly increasing sequences with quadratic growth $a_n\le Mn^2$ for $n\ge n_0$, $M\le1/2$; such a set has at least $\lfloor(N/M)^{1/2}\rfloor$ elements up to $N$ once that number is at least $n_0$, and $(N/M)^{1/2}\ge(2N)^{1/2}$, so the question is of the square-root density the problem asks about. The paper gives no answer.