Szemerédi theorem - Knowlpedia

Statement

If $E\subseteq\mathbb Z$ has positive upper Banach density, then for every $k\geq1$ there exist integers $a$ and $r\geq1$ such that

a,a+r,\ldots,a+kr\in E.

In particular, the conclusion holds for a subset of $\mathbb N$ with positive ordinary upper density. This is Szemerédi's theorem.

Ergodic proof route

Use the correspondence principle to obtain an event of positive probability. Multiple recurrence gives a positive intersection of its translates at $0,r,\ldots,kr$ . The correspondence inequality forces a nonempty intersection $E\cap(E-r)\cap\cdots\cap(E-kr)$ , and any element of it is the required starting point.

The common difference is positive, so the progression is nonconstant. Positive density is a sufficient hypothesis; many zero-density sets also contain long progressions.

References

Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. Theorem 1.1.8 and Lecture 10.