Another proposal for a polymath project is the following question of Michael Boshneritzan:
Question. Let be a (simple) path in a lattice which has bounded step sizes, i.e. for some C and all i. Is it necessarily the case that this path contains arbitrarily long arithmetic progressions, i.e. for each k there exists with r non-zero such that .
The d=1 case follows from Szemerédi’s theorem, as the path has positive density. Even for d=2 and k=3 the problem is open. The question looks intriguingly like the multidimensional Szemerédi theorem, but it is not obvious how to deduce it from that theorem. It is also tempting to try to invoke the Furstenberg correspondence principle to create an ergodic theory counterpart to this question, but to my knowledge this has not been done. There are also some faint resemblances to the angel problem that has recently been solved.
In honour of mini-polymath1, one could phrase this problem as a grasshopper trying to jump forever (with bounded jumps) in a square lattice without creating an arithmetic progression (of length three, say).
It is also worth trying to find a counterexample if C, d, k are large enough. Note that the continuous analogue of the problem is false: a convex curve in the plane, such as the parabola , contains no arithmetic progressions, but is rectifiable. However it is not obvious how to discretise this example.
In short, there seem to be a variety of tempting avenues to try to attack this problem; it may well be that many of them fail, but the reason for that failure should be instructive.
Andrew Mullhaupt informs me that this question would have applications to short pulse rejected Boolean delay equations.