Let n be a positive integer, and c a vector in Rn whose components are linearly independent over Q. Is it true that any line with direction vector c passes arbitrarily close to a point with integer coordinates?
This is true for n = 2, by the simplest result about approximating irrationals by rationals. Maybe it is too much to ask for it to hold in general; if so, I’d like a specific example and a specific counterexample.
This problem is connected to a question about homogeneous Cayley objects; I will explain the connection in the series on this topic.