Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On finite configurations in the spectra of singular measures

Rami Ayoush

Abstract

We establish various forms of the following certainty principle: a set $S \subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that $S$ is a support of the Fourier transform of a sufficiently singular probability measure on $\mathbb{R}^{n}$. As its main corollary, we provide new dimensional estimates for PDE- and Fourier-constrained vector measures. Those results, in certain cases of restrictions given by homogeneous operators, improve known bounds related to the notion of the $k$-wave cone.

On finite configurations in the spectra of singular measures

Abstract

We establish various forms of the following certainty principle: a set contains a given finite linear pattern, provided that is a support of the Fourier transform of a sufficiently singular probability measure on . As its main corollary, we provide new dimensional estimates for PDE- and Fourier-constrained vector measures. Those results, in certain cases of restrictions given by homogeneous operators, improve known bounds related to the notion of the -wave cone.

Paper Structure

This paper contains 9 sections, 14 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Applications of multilinear forms counting finite configurations
  4. Proof of Theorem \ref{['mainres2']}
  5. Other certainty principles and their connections with the Łaba-Pramanik theorem
  6. Sketches of proofs of Theorem \ref{['mainres']} and Theorem \ref{['mrquant']}
  7. Proof of Theorem \ref{['comb2']}
  8. Proof of Theorem \ref{['sharper']}
  9. Acknowledgements

Key Result

Theorem 1.4

Let $k\geq 1$ and let $B_{1}, \dots, B_{k} \in \mathop{\mathrm{Mat}}\nolimits_{n\times n}(\mathbb{R})$ be a sequence of $n \times n$ matrices. Suppose that $\mu \in \mathop{\mathrm{M}}\nolimits^{+}(\mathbb{R}^{n})$ is a finite, non-negative measure such that at some $x \in \mathbb{R}^{n}$ we have Then $\mathop{\mathrm{spec}}\nolimits(\mu)$ contains a non-trivial $\mathbb{B}$-configuration. Moreov

Figures (2)

  • Figure 1: A part of construction: surgeries near the points $(1,0,0)$ and $(-1,0,0)$.
  • Figure 2: The result of the application of $R$ ($\Gamma$ - grey, $R(\Gamma)$ - black).

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5: RW
  • Theorem 1.6: Theorem 3 in RW
  • Theorem 1.7: Theorem 1.3., Corollary 1.4. in ADHR
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 22 more