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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.
Item Metadata
Title |
Fractal uncertainty principles for ellipsephic sets
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2021
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Description |
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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Genre | |
Type | |
Language |
eng
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Date Available |
2021-04-23
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-ShareAlike 4.0 International
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DOI |
10.14288/1.0396939
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2021-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-ShareAlike 4.0 International