Rectifiable sets and the traveling salesman problem (Q807767)

The author shows that a bounded set K (\(\subset {\mathbb{C}})\) is contained in a rectifiable curve if and only if \[ \sum \{\frac{\omega (Q)}{\ell (Q)}\}^ 2 \ell (Q)<\infty. \] Here the summation is taken over all dyadic squares Q, \(\ell (Q)\) denotes the sidelength of \({\mathbb{Q}}\) and \(\omega\) (Q) is the width of an infinite strip \(S_ Q\) with smallest possible width such that \(S_ Q\supset K\cap 3Q\). This assertion is very deep and applicable to study various problems concerning harmonic measure and the Cauchy integral on curves.