Co-analytic Counterexamples to Marstrand’s Projection Theorem

Abstract: A recent “point-to-set principle" of J. Lutz and N. Lutz characterises the Hausdorff dimension of any subset of Euclidean space in terms of the complexity of its individual points. “Complexity" here refers to Kolmogorov complexity—so the point-to-set principle gives us an algorithmic handle on classical problems in fractal geometry: sets with particular fractal properties can now be constructed in stages, point-by-point, by coding “enough” information into each point, bit-by-bit. I will give a brief introduction to all these notions, and present a new result in fractal geometry whose proof uses such effective methods: under V=L, I will outline the construction of co-analytic counterexamples to Marstrand’s Projection Theorem, one of fractal geometry’s seminal theorems about the dimension of orthogonal projections of analytic plane sets onto lines. Our results also show that Marstrand’s theorem is indeed sharp for analytic sets, a fact previously unknown.

logic

Audience: researchers in the topic

Computability theory and applications

Series comments: Description: Computability theory, logic

The goal of this endeavor is to run a seminar on the platform Zoom on a weekly basis, perhaps with alternating time slots each of which covers at least three out of four of Europe, North America, Asia, and New Zealand/Australia. While the meetings are always scheduled for Tuesdays, the timezone varies, so please refer to the calendar on the website for details about individual seminars.