One World Fractals, 22 March 2023

• Vilma Orgoványi (video).
  • Overlapping random self-similar sets on the line.
  • Abstract: The Mandelbrot percolation fractal in \(\mathbb{R}^d\) is constructed inductively as follows. We fix an integer \(M\geq 2\) and a probability \(p\in(0,1)\). The closed unit cube is divided into \(M^d\) congruent sub-cubes, each of which is independently retained with probability \(p\) and discarded with probability \(1-p\). This process is repeated in the retained cubes ad infinitum, or until there are no cubes left. The Mandelbrot percolation process restricted to the building blocks of a sponge yields a random sponge. We consider the rational projections of random sponges into lines from the perspective of Hausdorff dimension, positivity of Lebesgue measure and existence of interior points. In contrast to Mandelbrot percolation, the inhomogeneity of the sponge yields qualitatively different phenomena.
    This is based on joint work with Károly Simon.
     
• Pablo Shmerkin (video).
  • Improved bounds for the dimensions of Furstenberg sets.
  • Abstract: An important and multi-faceted problem in geometric measure theory is understanding incidences between \(\delta\)-tubes and \(\delta\)-balls. The Furstenberg set problem is a concrete instance of this problem. I will present some \(\epsilon\)-improvements on the previously known bounds for Furstenberg sets, obtained in collaborations with T. Orponen and with H. Wang.
     
• Alec Chamberlain Cann (video).
  • More refined multifractal spectra for Bedford McMullen carpets.
  • Abstract: We study self-affine measures on Bedford McMullen carpets. The multifractal spectrum assesses the size of sets of points for which the local dimension takes a prescribed value. We make a refinement to the known spectrum that takes into account points for which the local dimension does not necessarily exist by considering upper and lower symbolic local dimension. For carpets on a \(n \times 2\) grid, and the measure that assigns mass uniformly on rectangles, we exhibit different behaviour from that seen when the refined spectrum is computed for self-similar sets.
     
• Organisers: Thomas Jordan and Henna Koivusalo

One World Fractals, 18 January 2023

• Amlan Banaji (video).
  • Dimensions of self-affine carpets.
  • Abstract: For self-affine carpets, different notions of fractal dimension such as Hausdorff, box and Assouad dimension often take different values. In the setting of Bedford-McMullen carpets, we will describe some of the methods used to calculate these dimensions, as well as two other dimension spectra, namely the Assouad spectrum and intermediate dimensions, which provide more refined local and global geometric information about the sets. For more general carpets of Lalley-Gatzouras type, the Assouad spectrum can display more interesting behaviour. This talk is based on joint work with Jonathan Fraser and István Kolossváry.
     
• Hong Wang (video).
  • Restricted projections in R^d.
  • Abstract: We prove a Marstrand-type projection theorem for directions in a non-degenerate curve. The proof uses Bourgain-Demeter and Bourgain-Demeter-Guth decoupling theorems. This is based on joint work with Gan and Guo.
     
• Organisers: Jonathan Fraser and Natalia Jurga

One World Fractals, 2 November 2022

• De-Jun Feng (video).
  • Dimensions of projected sets and measures on typical self-affine sets.
  • Abstract: Let \(\{T_ix+a_i\}_{i=1}^\ell\) be an iterated function system on \({\Bbb R}^d\) consisting of invertible affine maps with \(\|T_i\|<1/2\), and \(\pi:\{1,\ldots,\ell\}^{\Bbb N}\to {\Bbb R}^d\) the corresponding coding map. For every Borel set \(E\) and every Borel probability measure \(\mu\) in the coding space, we determine the various dimensions of their projections under \(\pi\) for typical translations \((a_1,\ldots, a_\ell)\); in particular, we give a necessary and sufficient condition on \(\mu\) so that the typical projection of \(\mu\) is exact dimensional. This extends the known results in the literature on typical projections of invariant sets and invariant measures. It plays an analogue to the classical theorems for fractal dimensions under orthogonal projections.
    The talk is based on joint work with Chiu-Hong Lo and Cai-Yun Ma.
     
• Aleksi Pyörälä (video).
  • On local structure of self-conformal measures.
  • Abstract: A fundamental concept in analysis is that of tangent. Tangents capture the local structure of an object but are often much more regular, and by studying the collection of tangents at every point one can obtain information on global properties of the object. A similar idea holds for measures: One can obtain very strong statements on global geometric properties of a measure by studying the sequences of measures, called "sceneries", obtained by "zooming in" around typical points for the measure. In our recent work with Balázs Bárány, Antti Käenmäki and Meng Wu, we study the sceneries of self-conformal measures in the absence of any separation conditions. It turns out that on the line, these measures are always uniformly scaling, meaning that the sceneries at typical points share similar statistics almost everywhere. As one corollary, we obtain that typical numbers for self-conformal measures are always normal in most bases.
     
• Amir Algom (video).
  • Fourier decay for self-similar measures.
  • Abstract: In recent years there has been an explosion of interest and progress on the Fourier decay problem for fractal measures. We will survey some of these results and remaining open problems, focusing on self-similar measures. We will also outline a method to tackle this problem for smooth images of the measure, when the IFS in question has non-cyclic contraction ratios.
    Joint work with Federico Rodriguez Hertz and Zhiren Wang.
     
• Organisers: Antti Käenmäki and Sascha Troscheit