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The Probability that a Convex Body Intersects the Integer Lattice in a <Emphasis Type="Italic">k</Emphasis>-dimensional Set
Authors:Edgardo Roldán-Pensado
Institution:1.Department of Mathematics,University College London,London,UK
Abstract: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|→∞.
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