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Statement

Setting (p. 9). For a primitive sequence $A$ (integers $0<a_1<a_2<\cdots$, no term dividing another), $f_A(x)=\sum_{a_i<x}1/a_i$; the constants $c_i$ of the paper are positive.

Theorem 2 (p. 15). Let $A$ be a primitive sequence and let $x_1,x_2,\ldots$ be any sequence with

$$\log\log x_{\nu+1}>(1+c_{17})\log\log x_\nu,\qquad(26)$$

where $c_{17}$ is an arbitrary constant. Put

$$\varepsilon_\nu=\frac{(\log\log x_\nu)^{1/2}}{\log x_\nu}\,f_A(x_\nu).$$

Then $\sum_{\nu=1}^\infty\varepsilon_\nu<c_{18}$, where $c_{18}$ depends only on $c_{17}$.

The paper introduces Theorem 2, at the foot of p. 14, as a sharpening of Theorem 1. Its hypothesis does not ask $A$ to be infinite, and $c_{18}$ does not depend on $A$ or on the sequence $x_\nu$.

No proof is given (p. 15). The paper says the proof is very similar to that of Theorem 1, that (26) can probably be much improved, and suggests that Theorem 2 may remain true with (26) replaced by $\log\log x_{\nu+1}>\log\log x_\nu+c_{19}(\log\log x_\nu)^{1/2}$; that weakening is put as a possibility, not proved.

Source. P. Erdős, A. Sárközy and E. Szemerédi, On a theorem of Behrend, J. Austral. Math. Soc. 7 (1967), 9--16: Theorem 2 and the remarks after it on p. 15. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the remark after it were read clause by clause on the printed page. The paper prints no proof, so none was checked. Nothing here is independently reviewed.

Proof pointer

None in the paper; it refers the reader to the proof of Theorem 1 (pp. 10--14), summarized on the Theorem 1 page.

Dependencies

The method of Theorem 1 of the same paper, by the paper's own account.

Bears on

• Problem 143: for a set of integers the problem's hypothesis is that no element divides another, and Theorem 2 gives a summable form of the estimate $f_A(x)=o(\log x/(\log\log x)^{1/2})$ of Theorem 1 along any sequence satisfying (26). Like Theorem 1, it bears only on the integer case, and it is stated without proof.