Ana de Orellana [WS1]: Finding 3APs in fractal sets

Speaker
Ana de Orellana, University of Jyväskylä

Abstract
Understanding which notions of “largeness” force a set to contain three-term arithmetic progressions goes back to a question of Erdős and Turán, answered for sets of positive upper density by Roth. However, largeness can mean different things in different settings. For sets of Lebesgue measure zero, the Hausdorff dimension alone never forces the existence of three-term arithmetic progressions. A result by Łaba and Pramanik gives the sufficient condition of supporting a measure with large enough Frostman exponent and with Fourier dimension greater than 2/3. In this talk I will present joint work with Ben Krause where we weaken Łaba-Pramanik’s Fourier decay assumption with the help of the Fourier spectrum, a family of dimensions that interpolate between the Fourier and Sobolev dimensions.