The Probability that a Convex Body Intersects the Integer Lattice in a <Emphasis Type="Italic">k</Emphasis>-dimensional Set

Let K be a convex body in ℝ ^d . It is known that there is a constant C ₀ depending only on d such that the probability that a random copy ρ(K) of K does not intersect ℤ ^d is smaller than \fracC₀|K|\frac{C_{0}}{|K|} and this is best possible. We show that for every k<d there is a constant C such that the probability that ρ(K) contains a subset of dimension k is smaller than \fracC|K|\frac{C}{|K|}. This is best possible if k=d−1. We conjecture that this is not best possible in the rest of the cases; if d=2 and k=0 then we can obtain better bounds. For d=2, we find the best possible value of C ₀ in the limit case when width(K)→0 and |K|→∞.