On the complexification of the Weierstrass non-differentiable function

It is shown that for the Weierstrass nowhere differentiable functions X a , b (t) = Σ ∞ n = 0 a n cos(b n t) and Y a , b (t) = Σ ∞ n = 0 a n sin(b n t) the set (X a , b , Y a , b ) ([0,2π]) has a non-empty interior in R 2 , provided b E N, b > 2 and a < 1 is sufficiently close to 1. It follows that the box dimension of graph(X a , b , Y a , b ) is equal to 3 - 2a where a = - log a/ log b and its Hausdorff dimension is at least 2. Moreover, the level sets L(s) for X a , b and Y a , b have Hausdorff dimension at least a for open sets of s E R, so the Hausdorff dimension of graph X a , b and graph Y a , b is at least 1 + a.