Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A combinatorial proof of a sumset conjecture of Furstenberg

Daniel Glasscock, Joel Moreira, Florian K. Richter

Abstract

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $\log r / \log s$ is irrational and $X$ and $Y$ are $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $λX + ηY$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.

A combinatorial proof of a sumset conjecture of Furstenberg

Abstract

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if is irrational and and are - and -invariant subsets of , respectively, then . Our main result yields information on the size of the sumset uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.

Paper Structure

This paper contains 18 sections, 21 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. History and context
  3. Overview of the paper
  4. Acknowledgements
  5. Preliminary definitions and results
  6. Continuous and discrete fractal geometry
  7. Multiplicatively invariant subsets of the reals and their finite approximations
  8. A quantitative equidistribution lemma
  9. A discrete Marstrand projection theorem
  10. A discrete projection theorem
  11. A corollary for oblique projections
  12. Trees and a subtree regularity theorem
  13. Preliminary definitions
  14. A subtree regularity theorem
  15. Proof of the sumsets theorem
  16. ...and 3 more sections

Key Result

Theorem 1

Let $r$ and $s$ be multiplicatively independent positive integers, and let $X, Y \subseteq [0,1]$ be $\times r$- and $\times s$-invariant sets, respectively. Define $\overline{\gamma} = \min ( \dim_{\text{H}} X + \dim_{\text{H}} Y, 1 )$. For all compact $I \subseteq \mathbb{R} \backslash \{0\}$ and

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 1.1: Marstrand_1954
  • Theorem 1.2: localentropy and shmerkinwu
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 45 more