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Fractal uncertainty principles for ellipsephic sets Hu, Nicholas

Abstract

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 δ for any given δ ∊ (½, 1), thereby confirming that the improvement over the basic exponent may be arbitrarily small for all δ ∊ (0, 1). Furthermore, using the theory of prolate matrices, we show that the exponents β₁ 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.

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