We show that if a setEin the positive real line has Hausdorff dimension greater thand/2m, then them-fold algebraic sum of the image ofEbyd-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 ofEis not close to being necessary