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Statement

Property DW is defined on the definitions page.

Theorem 8 (p. 10, quoted). "For any $\varepsilon>0$, there exists a sequence $A=\{a_n\}$ of positive integers such that $A$ does not have property $DW$ and, for all large $n$, $a_n<\exp(n^{\varepsilon})$."

The paper asks the reader to compare the remarks after Corollary 2, which give property DW when $a_n<e^{n^{\varepsilon}}$ for all large $n$ for every $\varepsilon>0$; here a single $\varepsilon$ is fixed.

Source. Brown, T. C., Erdős, P. and Freedman, A. R., Quasi-progressions and descending waves, J. Combin. Theory Ser. A 53 (1990), no. 1, 81--95, doi:10.1016/0097-3165(90)90021-N, read in the authors' copy identified on the source card, whose pages are numbered 1 to 13: the statement on p. 10, the proof on pp. 10--11.

Read depth. Claims checked: the statement was read on the print's page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 4, pp. 10--11. Let $A^N$ be the set of sums of $N$ distinct powers of 2. Counting such sums below $2^{i}$ shows that the $n$th element of $A^N$ is less than $\exp(n^{\delta})$ for large $n$ whenever $\delta>1/N$, so one takes $N>1/\varepsilon$. That $A^N$ has no arbitrarily long descending waves is proved by induction on $N$, starting from the powers of 2, which contain no 3-term descending wave: in a long wave the leading binary exponent must increase many times, and then a later gap exceeds the first.

Dependencies

None outside the paper.

Bears on

No problem page.