LARGE SETS CONTAINING NO COPIES OF A GIVEN INFINITE SEQUENCE

Let A be a discrete, unbounded, infinite set in R. Can we find a "large" measurable set E in R which does not contain any affine copy x + tA of A (for any in R, t > 0)? If a(n) is a real, nonnegative sequence that does not increase ex- ponentially, 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 gen- eralize this to higher dimensions and also for some "non-linear" copies of the sequence. Our method is probabilistic. Joint work with M. Kolountzakis (Univ. of Crete). Current address: Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, Greece