Analyst's traveling salesman theorem

A posteriori, for E to be contained in a rectifiable curve Γ, since Γ has tangents at H¹-almost every point in Γ (where H¹ denotes one-dimensional Hausdorff measure), E must look flat when you zoom in on points in E. This suggests that a condition that would tell us whether a set could be contained in a curve must somehow incorporate information about how flat E is when we zoom in on points of E at different scales.

This discussion motivates the definition of the following quantity:

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

Where Q is any square, ${\displaystyle \ell (Q)}$ is the sidelength of Q, and dist(x, L) measures the distance from x to the line L. Intuitively, ${\displaystyle 2\beta _{E}(Q)\ell (Q)}$ is the width of the smallest rectangle containing the portion of E inside Q, and hence ${\displaystyle \beta _{E}(Q)}$ gives us 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. 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).