Analyst's traveling salesman theorem

A rectifiable curve has tangents at almost all of its points, where in this case "almost all" means all but a subset whose one-dimensional Hausdorff measure is zero. Accordingly, if a set is contained in a rectifiable curve, the set must look flat when zooming in on almost all of its points. This suggests that testing us whether a set could be contained in a rectifiable curve must somehow incorporate information about how flat it is when one zooms in on its points at different scales.

This discussion motivates the definition of the following quantity, for a plane set ${\displaystyle E\subset \mathbb {R} ^{2}}$:

${\displaystyle \beta _{E}(Q)={\frac {1}{\ell (Q)}}\inf\{\delta$ :{}} there is a line ${\displaystyle L}$ so that for every ${\displaystyle x\in E\cap Q,}$ dist${\displaystyle (x,L)<\delta \},}$

where ${\displaystyle E}$ is the set that is to be contained in a rectifiable curve, ${\displaystyle Q}$ is any square, ${\displaystyle \ell (Q)}$ is the side length of ${\displaystyle Q}$, and dist${\displaystyle (x,L)}$ measures the distance from ${\displaystyle x}$ to the line ${\displaystyle L}$. Intuitively, ${\displaystyle 2\beta _{E}(Q)\ell (Q)}$ is the width of the smallest rectangle containing the portion of ${\displaystyle E}$ inside ${\displaystyle Q}$, and hence ${\displaystyle \beta _{E}(Q)}$ gives a scale invariant notion of flatness.

Let Δ denote the collection of dyadic squares, that is,

${\displaystyle \Delta =\{[i2^{k},(i+1)2^{k}]\times [j2^{k},(j+1)2^{k}]:i,j,k\in \mathbb {Z} \},}$

where ${\displaystyle \mathbb {Z} }$ denotes the set of integers. For a set ${\displaystyle E\subseteq \mathbb {R} ^{2}}$, define

${\displaystyle \beta (E)={\text{diam}}E+\sum _{Q\in \Delta }\beta _{E}(3Q)^{2}\ell (Q)}$

where diam E is the diameter of E and ${\displaystyle 3Q}$ is the square with same center as ${\displaystyle Q}$ with side length ${\displaystyle 3\ell (Q)}$. Then Peter Jones's[1] analyst's traveling salesman theorem may be stated as follows:

• There is a number C > 0 such that whenever E is a set with such that β(E) < ∞, E can be contained in a curve with length no more than Cβ(E).
• Conversely (and substantially more difficult to prove), if Γ is a rectifiable curve, then β(Γ) < CH¹(Γ).