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Prescribed projections and efficient coverings by curves in the plane

Alan Chang, Alex McDonald, Krystal Taylor

TL;DR

This work extends Davies' efficient covering to a nonlinear setting by using translates of a fixed curve $\Gamma$ to efficiently cover a plane-measurable set $W$ up to measure zero. The authors introduce a nonlinear prescribed projection framework via a 1-parameter family of generalized projections $\{\Phi_\alpha\}$ associated to $\Gamma$, and solve the problem using a novel Venetian blind construction that combines small-projection control with covering along curves. A key local nonlinear key lemma, proven by polygonal approximation and iterated blinds, serves as the foundation for constructing a global covering $E$ such that $E+\Gamma$ covers $W$ with zero excess. The results connect to Kakeya/Nikodym-type phenomena and provide a nonlinear analogue of Falconer’s prescribed projection theorem, with implications for geometric measure theory and harmonic analysis of curve-saturating coverings.

Abstract

Davies efficient covering theorem states that an arbitrary measurable set $W$ in the plane can be covered by full lines so that the measure of the union of the lines has the same measure as $W$. This result has an interesting dual formulation in the form of a prescribed projection theorem. In this paper, we formulate each of these results in a nonlinear setting and consider some applications. In particular, given a measurable set $W$ and a curve $Γ=\{(t,f(t)): t\in [a,b]\}$, where $f$ is $C^1$ with strictly monotone derivative, we show that $W$ can be covered by translations of $Γ$ in such a way that the union of the translated curves has the same measure as $W$. This is achieved by proving an equivalent prescribed generalized projection result, which relies on a Venetian blind construction.

Prescribed projections and efficient coverings by curves in the plane

TL;DR

This work extends Davies' efficient covering to a nonlinear setting by using translates of a fixed curve to efficiently cover a plane-measurable set up to measure zero. The authors introduce a nonlinear prescribed projection framework via a 1-parameter family of generalized projections associated to , and solve the problem using a novel Venetian blind construction that combines small-projection control with covering along curves. A key local nonlinear key lemma, proven by polygonal approximation and iterated blinds, serves as the foundation for constructing a global covering such that covers with zero excess. The results connect to Kakeya/Nikodym-type phenomena and provide a nonlinear analogue of Falconer’s prescribed projection theorem, with implications for geometric measure theory and harmonic analysis of curve-saturating coverings.

Abstract

Davies efficient covering theorem states that an arbitrary measurable set in the plane can be covered by full lines so that the measure of the union of the lines has the same measure as . This result has an interesting dual formulation in the form of a prescribed projection theorem. In this paper, we formulate each of these results in a nonlinear setting and consider some applications. In particular, given a measurable set and a curve , where is with strictly monotone derivative, we show that can be covered by translations of in such a way that the union of the translated curves has the same measure as . This is achieved by proving an equivalent prescribed generalized projection result, which relies on a Venetian blind construction.
Paper Structure (13 sections, 18 theorems, 89 equations, 9 figures)

This paper contains 13 sections, 18 theorems, 89 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation and elementary calculations
  3. Outline of the proof strategy
  4. Acknowledgments
  5. The Venetian blind construction
  6. Dividing
  7. Rotating
  8. Venetian blinds
  9. Iterated Venetian blinds
  10. Key lemma for prescribed projections
  11. Proof of main theorem
  12. Appendix: Proof of parabola example in Remark \ref{['general_classes']}
  13. Appendix: Duality and the equivalence of Davies' theorem and the prescribed projection theorem

Key Result

Theorem 1.1

Let $W\subset \mathbb{R}^2$ be measurable. Let $f:[a,b]\to\mathbb{R}$ be a $C^1$ function with a strictly monotone derivative, and let $\Gamma \subset \mathbb{R}^2$ be the graph of $f$. Then there exists a set $E\subset \mathbb{R}^2$ such that

Figures (9)

  • Figure 1.1: Projections of a horizontal segment
  • Figure 1.2: Extending the "small projection" range
  • Figure 1.3: Venetian blinds
  • Figure 1.4: Fibers of curve projections with tangents in covering directions and small directions.
  • Figure 1.5: Venetian blinds and fibers of curve projections
  • ...and 4 more figures

Theorems & Definitions (42)

  • Theorem 1.1: Efficient covering by curves
  • Theorem 1.2: Non-linear prescribed projection theorem
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1: Small projections in certain directions
  • proof
  • Lemma 2.2: Basic geometry of $\mathsf{ROTATE}$
  • proof
  • ...and 32 more