The Dimension of the Brownian Frontier Is Greater Than 1

Consider a planar Brownian motion run for finite time. Thefrontieror “outer boundary” of the path is the boundary of the unbounded component of the complement. K. Burdzy (Prob. Theory Related Field83(1989), 135–205) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension 4/3, but this is still open.) The proof uses Jonesʹs Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands.