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Statement

Setting (pp. 447--449). $P=\{1,2,3,\ldots\}$; $\langle\rho:1\rangle$ is the least subset of $P$ containing $1$ and closed under the operation $\rho$; a per-set is a finite union of infinite arithmetic progressions in $P$.

Conjecture 1 (p. 457, quoted). "Suppose $r,m_1,\cdots,m_r\in P$, and $(m_1,\cdots,m_r)=1$. Then $\langle m_1x_1+\cdots+m_rx_r:1\rangle$ is a per-set."

Status in the paper

The paper says all the results of its Section 3 were motivated by attempts to prove this conjecture (p. 457). It notes that, for $r\ge3$, Theorem 10 (p. 459) passes the per-set property from $\langle m_1x_1+\cdots+m_{r-1}x_{r-1}:1\rangle$, when $(m_1,\ldots,m_{r-1})=1$, to $\langle m_1x_1+\cdots+m_rx_r:1\rangle$, and that most of its effort went into the case $\langle mx+ny:1\rangle$ with $(m,n)=1$, where it proves Theorem 11 for $m=2$ and odd $n$. It adds, in parentheses on p. 457, that Conjecture 1 has been proved, citing the authors' paper "Sets generated by iteration of a linear operation", then to appear in Pacific J. Math. (its reference [2]). That proof is not in this paper.

Proof pointer

None in this paper; the paper attributes the proof to its reference [2].

Read depth

Claims checked: the conjecture and the surrounding remarks on p. 457 were read on the page images of the print. The proof reported in reference [2] was not read. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. D. A. Klarner and R. Rado, Arithmetic properties of certain recursively defined sets, Pacific J. Math. 53 (1974), no. 2, 445--463, doi:10.2140/pjm.1974.53.445; the edition read is named on the source card.

Bears on

None of the problem pages directly.