Morris: Unavoidable structures in positive-density subsets of Euclidean space; or, how I learned to stop worrying and love the weak-* topology on L^∞(ℝ^d)
Abstract:
Inspired by a question of L. A. Szekely, in the early 1980s Furstenberg, Katznelson and Weiss proved the following theorem: if a measurable subset of the plane has positive density in a certain natural sense, then for all sufficiently large real numbers D we may find a pair of points in the set which are separated by Euclidean distance D. Extensions and alternative proofs were subsequently given by various authors including Bourgain, Falconer-Marstrand, Quas, and Bukh. We prove a general sufficient condition for a property to hold at all sufficiently large scales in all positive-density subsets of ℝ^d, improving a theorem of B. Bukh. Our proof uses a characterisation of the density of a measurable set in terms of the translation orbit of its characteristic function, which we view as an element of L^∞(ℝ^d) equipped with the weak-* topology. The result is then deduced using a pointwise ergodic theorem for ℝ^d-actions. Using this method we give new proofs of theorems of Furstenberg-Katznelson-Weiss, Marstrand, and R. L. Graham.