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Mathematics > Classical Analysis and ODEs
Title: Fractal uncertainty principle for random Cantor sets
(Submitted on 23 Apr 2024)
Abstract: We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel (arXiv:2107.08276), we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension d is in (0,2/3). We proved that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds for these discrete Cantor sets with random alphabets.
In this sequel, we construct random Cantor sets with dimension d in (0,2/3) in R via a different random procedure from the one in the prequel. We prove that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.
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