Kahane using brownian local times of intersection

We show that if a set E in the positive real line has Hausdorff dimension greater than d/2 m, then the m-fold algebraic sum of the image of E by d-dimensional Brownian motion has an interior point. This extends a result of Kahane. The proof uses techniques found in Rosen (1983) and Geman, Horowitz and Rosen. We then show that the results do not hold for random sets and demonstrate that the above condition on the Hausdorff dimension of E is not close to being necessary