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Quantitative visibility estimates for unrectifiable sets in the plane

M. Bond, I. Laba, J. Zahl

Abstract

The "visibility" of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak (Real Anal. Exchange 2006/07) proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of $δ$-neighbourhoods of such sets. We also prove lower bounds on the visibility of $δ$-neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates

Quantitative visibility estimates for unrectifiable sets in the plane

Abstract

The "visibility" of a planar set from a point is defined as the normalized size of the radial projection of from to the unit circle centered at . Simon and Solomyak (Real Anal. Exchange 2006/07) proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of -neighbourhoods of such sets. We also prove lower bounds on the visibility of -neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates

Paper Structure

This paper contains 29 sections, 38 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Upper bounds for self-similar sets
  3. Lower bounds on visibility
  4. Acknowledgements
  5. Warm-up results
  6. Notation
  7. Visibility and Favard length
  8. Energy methods
  9. Visibility dimension of self-similar sets
  10. Visibility lower bounds
  11. Initial reductions and discretization
  12. Discretization of the points
  13. Discretization of the lines
  14. Discretization of visibility
  15. Separating the vantage points from the set $A$
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $\mathcal{J}$ be a 1-dimensional self-similar set defined by homotheties with equal contraction ratios. Then:

Figures (3)

  • Figure 1: The disks $Q_1,...,Q_6\in \mathcal{Q}_L$ form a tall stack.
  • Figure 2: Continuing from Figure \ref{['greenL']}, this is a rough partial sketch of $\mathcal{J}_M$, $M>L$. $T_w(Q_1),...,T_w(Q_6)$ are singled out, for some $w\in W_{M-L}$.
  • Figure 3: A closer look at a smaller portion of $\mathcal{J}_n$, $n>M>L$. Since $T_w(Q_4)<q$, it follows that $Q_4\prec q$. In words, "$q$ is a descendant of a self-similar copy of $Q_4$."

Theorems & Definitions (76)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • ...and 66 more