The fractal dimension is \(\frac{\log{2}}{\log{3}}\). After having done the removal process \(n\) times, there are \(2^n\) intervals, each of length \(3^{-n}\), so the fractal dimension is
\[ \lim_{\varepsilon \rightarrow 0} -\frac{\log{N(\varepsilon)}}{\log{(\varepsilon)}} = \lim_{n \rightarrow \infty} -\frac{\log{2^{n}}}{\log{3^{-n}}} = \lim_{n \rightarrow \infty} \frac{n\log{2}}{n\log{3}} = \frac{\log{2}}{\log{3}}. \]