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When is the fractal uncertainty principle for discrete Cantor sets most uncertain?

Chun-Kit Lai, Ruxi Shi

TL;DR

The work identifies distributed spectral pairs as the exact mechanism to attain the most uncertain exponent in the discrete fractal uncertainty principle for Cantor sets. It proves a precise equivalence: β(M,A,B) equals (1−δ)/2 if and only if (A,B) forms a distributed spectral pair, and it provides a constructive, inductive proof along with a complementary self-similar-measure framework. The paper also advances the theory by classifying distributed spectral pairs in several cyclic groups, including Z_{M^2} and key cases for M with prime-factor structure, unveiling when maximal uncertainty is possible and when it is not. In addition, it discusses extensions to continuous FUP settings and outlines open problems and conjectures, linking discrete spectral phenomena to Fuglede-type questions and self-similar measures. These results deepen the understanding of the spectral-tiling structure underlying maximal uncertainty and may guide future investigations in higher-dimensional and continuous analogues.

Abstract

We give a necessary and sufficient condition to achieve the most uncertain exponent in the fractal uncertainty principle of discrete Cantor sets. The condition will be described as distributed spectral pairs, which is a generalization of the spectral pair studied in the spectral sets literature. We investigate distributed spectral pairs in some cyclic groups and some complete classifications are given. Finally, we also discuss the most uncertain case in the continuous setting.

When is the fractal uncertainty principle for discrete Cantor sets most uncertain?

TL;DR

The work identifies distributed spectral pairs as the exact mechanism to attain the most uncertain exponent in the discrete fractal uncertainty principle for Cantor sets. It proves a precise equivalence: β(M,A,B) equals (1−δ)/2 if and only if (A,B) forms a distributed spectral pair, and it provides a constructive, inductive proof along with a complementary self-similar-measure framework. The paper also advances the theory by classifying distributed spectral pairs in several cyclic groups, including Z_{M^2} and key cases for M with prime-factor structure, unveiling when maximal uncertainty is possible and when it is not. In addition, it discusses extensions to continuous FUP settings and outlines open problems and conjectures, linking discrete spectral phenomena to Fuglede-type questions and self-similar measures. These results deepen the understanding of the spectral-tiling structure underlying maximal uncertainty and may guide future investigations in higher-dimensional and continuous analogues.

Abstract

We give a necessary and sufficient condition to achieve the most uncertain exponent in the fractal uncertainty principle of discrete Cantor sets. The condition will be described as distributed spectral pairs, which is a generalization of the spectral pair studied in the spectral sets literature. We investigate distributed spectral pairs in some cyclic groups and some complete classifications are given. Finally, we also discuss the most uncertain case in the continuous setting.
Paper Structure (15 sections, 16 theorems, 103 equations)

This paper contains 15 sections, 16 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The most uncertain exponent
  3. Main results
  4. Overview of the paper.
  5. Proof of Theorem \ref{['maintheorem1']}
  6. Proof of Theorem \ref{['maintheorem11']}(1)
  7. Proof of Theorem \ref{['maintheorem11']}(2).
  8. Associated self-similar measures.
  9. Cardinality of $\Omega_{q,k,L}$.
  10. Completion of the proof.
  11. Spectral pair in ${\mathbb Z}_{M^2}$
  12. Classification of distributed spectral pair in some cyclic groups
  13. Discussions and Open problems
  14. Discrete settings.
  15. Continuous settings

Key Result

Theorem 1.1

Let ${\mathcal{A}},{\mathcal{B}}\subset \{0,1,...,M-1\}$. Then there exists $\beta =\beta (M,{\mathcal{A}},{\mathcal{B}})> \max \left(0,\frac{1}{2}-\delta\right)$ such that

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • ...and 21 more