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Van der Corput and metric theorems for geometric progressions for self-similar measures

Amir Algom, Yuanyang Chang, Meng Wu, Yu-Liang Wu

Abstract

We prove a van der Corput lemma for non-atomic self-similar measures $μ$. As an application, we show that the correlations of all finite orders of $( x^n \mod 1 )_{n\geq 1}$ converge to the Poissonian model for $μ$-a.e. $x$, assuming $x>1$. We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay.

Van der Corput and metric theorems for geometric progressions for self-similar measures

Abstract

We prove a van der Corput lemma for non-atomic self-similar measures . As an application, we show that the correlations of all finite orders of converge to the Poissonian model for -a.e. , assuming . We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay.
Paper Structure (16 sections, 21 theorems, 146 equations)

This paper contains 16 sections, 21 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. A van der Corput lemma for fractal measures
  3. Applications to higher order correlations of sequences modulo 1
  4. Applications to Fourier decay of self-conformal measures with respect to real analytic IFSs
  5. Main technical Theorem: van der Corput for phases with non-flat derivatives
  6. Outline of method
  7. Acknowledgments
  8. Preliminaries
  9. Some properties of self-similar measures
  10. On Tsujii's large deviations estimate
  11. Non-affine continuously differentiable functions are non-flat
  12. Proof of Theorem \ref{['thm:van_der_Corput_C1+a']}
  13. Proof of Theorem \ref{['Main Theorem pos']}
  14. A covering lemma for a class of polynomials
  15. Proof of Theorem \ref{['main-2']}
  16. ...and 1 more sections

Key Result

Proposition 1

Stein1993bookLet $g$ be a real-valued smooth function on an interval $J \subset\mathbb{R}$. Suppose $|g^{(k)}(x)|\geq 1$ for all $x\in J$, where $k\geq 1$ is an integer. If $k=1$ and $g'$ is monotonic, or if $k>1$, then

Theorems & Definitions (32)

  • Proposition
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 22 more