FRACTAL UNCERTAINTY PRINCIPLE FOR RANDOM CANTOR SETS

Abstract We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30 (6) (2023), 1657–1679], we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension δ ∈ ( 0 , 2 3 ) $\delta \in (0,\frac 23)$ delta element of left parenthesis 0 comma two thirds right parenthesis . We proved that, with overwhelming probability, the FUP with an exponent ≥ 1 2 − 3 4 δ − ε $\ge \frac 12-\frac 34\delta -\varepsilon $ greater than or equals one half minus three fourths delta minus epsilon holds for these discrete Cantor sets with random alphabets. In this paper, we construct random Cantor sets with dimension δ ∈ ( 0 , 2 3 ) $\delta \in (0,\frac 23)$ delta element of left parenthesis 0 comma two thirds right parenthesis in R $\mathbb {R}$ double struck upper R via a different random procedure from the previous one used in Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30 (6) (2023), 1657–1679]. We prove that, with overwhelming probability, the FUP with an exponent ≥ 1 2 − 3 4 δ − ε $\ge \frac 12-\frac 34\delta -\varepsilon $ greater than or equals one half minus three fourths delta minus epsilon 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.