Morris talk at ergodic theory meeting

Ian Morris gave an invited talk at the One Day Ergodic Theory Meeting, held at the School of Mathematical Sciences, Queen Mary College, University of London, on Wednesday 7th November.  The talk was on “Unavoidable structures in positive-density subsets of Euclidean space; or, how I learned to stop worrying and love the weak-* topology on L-infinity“.  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 ℝd, improving a theorem of B. Bukh. Our proof uses a characterization 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(ℝd) equipped with the weak-* topology. The result is then deduced using a pointwise ergodic theorem for ℝd-actions. Using this method we give new proofs of theorems of Furstenberg-Katznelson-Weiss, Marstrand, and R. L. Graham.