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Statement

The paper defines the following notions for a finite increasing sequence $x_1<x_2<\cdots<x_k$ (pp. 1--2).

• Cube (p. 1, display (1)). For an integer $a$ and generators $y_1,\ldots,y_m$, the $m$-cube $\langle a,y_1,\ldots,y_m\rangle$ is the set of sums $a+\varepsilon_1y_1+\cdots+\varepsilon_my_m$ with every $\varepsilon_j\in\{0,1\}$.
• Quasi-progression (p. 2). The sequence is a $k$-term quasi-progression of diameter $d$, written $k-QP(d)$, when the set of its consecutive differences $x_{i+1}-x_i$, $1\le i\le k-1$, has diameter at most $d$; equivalently, some $N$ satisfies $N\le x_{i+1}-x_i\le N+d$ for $1\le i\le k-1$. A $k$-term arithmetic progression is a $k-QP(0)$.
• Combinatorial progression (p. 2). The sequence is a $k$-term combinatorial progression of order $d$, written $k-CP(d)$, when the integer parts $[x_{i+1}-x_i]$, $1\le i\le k-1$, take at most $d$ distinct values. A $k-QP(d)$ of integers is a $k-CP(d+1)$.
• Descending wave (p. 2). The sequence is a $k$-term descending wave, written $k-DW$, when its differences are non-increasing: $x_{j+1}-x_j\ge x_{j+2}-x_{j+1}$ for $1\le j\le k-2$.

A set of positive integers has property

• AP if it contains arbitrarily long arithmetic progressions, and C if it contains arbitrarily large cubes (p. 1);
• QP if, for some fixed $d$, it contains a $k-QP(d)$ for each $k\ge1$ (p. 2);
• CP if, for some fixed $d$, it contains a $k-CP(d)$ for each $k\ge1$ (p. 2);
• DW if it contains arbitrarily large descending waves (p. 2).

The paper notes (p. 2) that the sequence definitions apply to real sequences as well, and states without proof that a set of reals $R=\{r_1<r_2<\cdots\}$ with $r_{i+1}-r_i\ge1$ for all sufficiently large $i$ has property QP, CP or DW exactly when the set of integers $\{[r_i]:i\ge1\}$ has the same property.

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: cubes and the properties AP and C on p. 1, the remaining definitions and the remark on real sequences on p. 2.

Read depth. Claims checked: each definition was read clause by clause on the print's pages. Nothing here is independently reviewed.

Proof pointer

Definitions; the two implications noted above ($AP\Rightarrow QP$ and $QP\Rightarrow CP$) are immediate and are the first steps of Theorem 1.

Dependencies

None.

Bears on

• Problem 781: the problem's descending wave, $x_j\ge(x_{j+1}+x_{j-1})/2$ for $1<j<k$, is the paper's $k-DW$ rewritten, since the inequality says $x_j-x_{j-1}\ge x_{j+1}-x_j$ (an observation of this page).
• Problem 782: the problem's first question, whether some constant $C$ allows $k$-term sequences of squares with $x_i+d\le x_{i+1}\le x_i+d+C$ for every $k$, asks whether the squares have property QP with diameter $C$; its second asks whether they contain arbitrarily large cubes, that is, have property C (an observation of this page). The paper poses both questions in Section 5.