[Paper Review] A doubling measure on $\R^d$ can charge a rectifiable curve

John B. Garnett, Rowan Killip+1 more|ArXiv.org|Jun 13, 2009

Advanced Mathematical Modeling in Engineering3 references4 citations

TL;DR

This paper constructs a doubling measure on $ ^d$ ($d \geq 2$) that assigns positive measure to a rectifiable curve, contradicting the long-standing belief that doubling measures must be singular to rectifiable sets. The construction uses a product of one-dimensional doubling measures on $ $, built via a ternary-based iterative process with controlled mass distribution, and embeds a curve through a nested family of dyadic cubes, proving both finite $ ^1$-Hausdorff measure and positive measure under the doubling measure.

ABSTRACT

For $d\geq 2$, we construct a doubling measure $ν$ on $\R^d$ and a rectifiable curve $Γ$ such that $ν(Γ)>0$.

Motivation & Objective

• To resolve the open question of whether a doubling measure on $ ^d$ ($d \geq 2$) can assign positive measure to a rectifiable curve.
• To challenge the prevailing intuition that doubling measures must be singular to rectifiable sets and smooth submanifolds.
• To construct an explicit example of a doubling measure and a rectifiable curve where the measure charges the curve positively.
• To demonstrate that such a construction is possible despite the measure's doubling condition and the curve's finite length.

Proposed method

• Define a one-dimensional doubling measure $\mu$ on $\r$ using a product of functions based on ternary expansions, with mass distributed via a parameter $\delta \in (0, \frac{1}{3}]$.
• Construct the $d$-dimensional doubling measure $\nu$ as the $d$-fold product of $\mu$, ensuring the doubling property is preserved under product structure.
• Define a nested family of dyadic cubes $\mathcal{K}_l$ through iterative replacement of cubes with subcubes based on digit counts in ternary expansions.
• Build the curve $\Gamma$ as the union of skeleton paths within the cubes at each level and the limit set of the construction, ensuring connectedness and rectifiability.
• Use large deviation estimates to bound the measure of the set of cubes with at most $k$ non-1 digits in their first $n$ ternary digits.
• Apply the dominated convergence theorem and exponential bounds to estimate the lower bound of $\nu(\Gamma)$, showing it is positive for suitable parameters.

Experimental results

Research questions

• RQ1 Can a doubling measure on $\r^d$ ($d \geq 2$) assign positive measure to a rectifiable curve?
• RQ2 Is it possible to construct a doubling measure that is not singular to a rectifiable curve, despite the measure's regularity?
• RQ3 What structural constraints on the measure and curve allow such a construction to succeed?
• RQ4 How can the mass distribution in a doubling measure be tuned to concentrate on a set of finite $\mathcal{H}^1$-measure?

Key findings

• The paper constructs a doubling measure $\nu$ on $\r^d$ and a rectifiable curve $\Gamma$ such that $\nu(\Gamma) > 0$, resolving the open problem affirmatively.
• The curve $\Gamma$ has finite $\mathcal{H}^1$-measure, as shown by the bound $\mathcal{H}^1(\Gamma) \leq 3d2^d e^{3dn_1[\delta + \delta\log(\delta^{-1})]}$ under the parameter constraints.
• The measure $\nu(\Gamma)$ is bounded below by $\exp\bigl{\{} -\frac{de^{-2\delta^2n_1}}{(1-e^{-2\delta^2n_1})^2}\bigr{\}}$, which is positive for any fixed $\delta > 0$ and large $n_1$.
• The construction ensures that $\nu(\Gamma)$ can be made arbitrarily close to $\nu([0,1]^d)$ by increasing $n_1$, showing the curve can capture a large proportion of the measure.
• The method relies on a ternary-based iterative mass distribution where intervals with fewer 1-digits in their ternary expansion receive higher mass, controlled by $\delta$.