Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset [0,1]$ and $Y\subset [a,a+h^{-1}]$ are$\delta$-regular on the relevant scales, then, for $a\ge a_0{h}^{-1}$, $$\mathrm{supp}{\mathcal{H}}_{\nu }f\subset Y⟹\mathrm{\parallel }{\mathbf{1}}_{X}f{\mathrm{\parallel }}_{L_{\nu }^2}\le C{h}^{\beta }\mathrm{\parallel }f{\mathrm{\parallel }}_{L_{\nu }^2}\mathrm{.}$$ The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.