Fractal uncertainty principles (FUPs) in harmonic analysis quantify the extent to which a function and its Fourier transform can be simultaneously localized near a fractal set. We investigate the formulation of such principles for ellipsephic sets, discrete Cantor-like sets consisting of integers in a given base with digits in a specified alphabet. We employ a combination of theoretical and numerical methods to find and support our results.
To wit, we resolve a conjecture of Dyatlov and Jin by constructing a sequence of base-alphabet pairs whose FUP exponents converge to the basic exponent and whose dimensions converge to $\delta$ for any given $\delta \in (\frac{1}{2}, 1)$, thereby confirming that the improvement over the basic exponent may be arbitrarily small for all $\delta \in (0, 1)$. Furthermore, using the theory of prolate matrices, we show that the exponents $\beta_1$ of the same sequence decay subexponentially in the base.
In addition, we explore extensions of our work to higher-order ellipsephic sets using blocking strategies and tensor power approximations. We also discuss the connection between discrete spectral sets and base-alphabet pairs achieving the maximal FUP exponent.