Editing Analytic capacity (section)

===Positive length but zero analytic capacity===
Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of '''C''' and its analytic capacity, it might be conjectured that ''γ''(''K'') = 0 implies ''H''<sup>1</sup>(''K'') = 0. However, this conjecture is false. A counterexample was first given by [[Anatoli Georgievich Vitushkin|A. G. Vitushkin]], and a much simpler one by [[John B. Garnett]] in his 1970 paper. This latter example is the '''linear four corners Cantor set''', constructed as follows:

Let ''K''<sub>0</sub> := [0, 1] × [0, 1] be the unit square. Then, ''K''<sub>1</sub> is the union of 4 squares of side length 1/4 and these squares are located in the corners of ''K''<sub>0</sub>. In general, ''K<sub>n</sub>'' is the union of 4<sup>''n''</sup> squares (denoted by <math>Q_n^j</math>) of side length 4<sup>−''n''</sup>, each <math>Q_n^j</math> being in the corner of some <math>Q_{n-1}^k</math>. Take ''K'' to be the intersection of all ''K''<sub>''n''</sub> then <math>H^1(K)=\sqrt{2}</math> but ''γ''(''K'') = 0.