A “large scale” analogue of the Erdős similarity problem can be stated as follows: let A be a discrete, unbounded, infinite set in R; can we find a “large” measurable set E ⊂ R which does not contain any affine copy x+tA of A (for any x ∈ R, t > 0)? If an is a real, nonnegative sequence that does not increase exponentially, then, for any 0 ≤ p < 1, we construct a Lebesgue measurable set which has measure at least p in any unit interval and which contains no affine copy of the given sequence. We generalize this to higher dimensions and also for some “non-linear” copies of the sequence. Our method is probabilistic.
Joint work with M. Kolountzakis (University of Crete, Greece).
