Cubes, side lengths and centres and their Furstenberg type siblings

Fractal Geometry and Dynamics

30 November 15:00 - 15:50

Han Yu - University of St Andrews

We will discuss Bourgain-Wolff-Schlag circle maximal operators to see what happens if we replace circles by cubes or polygonal shapes. In recent work by (Keleti, Nagy, Shmerkin) and (Chang, Csörnyei, Héra and Keleti) (rotated)cubes with large centre set were considered. In this talk we talk about cubes with large side length set and similar result holds in that case for example, in dimension n, a set which contains surfaces of cubes with all side lengths has Hausdorff dimension at least n-1 and box/Assouad dimension n-0.5, this bound is sharp. This is not the end of the story. We show such sets must be large in general in a precise sense that will be formulated in the talk. As a by-product we will encounter a Kakeya type construction which we shall call 'grass sets', a corresponding 'grass conjecture' will be formulated. Everything above has Furtenberg type generalization which we will briefly discuss. As a by-product, we revisit Wolff's circle maximal inequality and a $\alpha,\beta$-Wolff problem will be formulated as a philosophical analogy of $\alpha,\beta$ Furstenberg problem. To be more related with Furtenberg sets, we discuss a certain dynamical defined set which has self-similar centre set and 'self-similar' way of assigning radius we shall see that under some rational independence such set must hit the expected large Assouad dimension. This can be seen as the Furtenberg times 2 times 3 set conjecture which was proved by Shmerkin and Wu independently. We conjectured that all other dimensions (Hausdorff for example) must hit the large expected value. Subject to time limit, only a subset of the above results which has 0 dimension can be presented
Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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