On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies

We determine the Hausdorff and box dimension of the fractal graphs of some Weierstrass- type functions of the form f (x) = Sigma(infinity)(n=1) a(n) g(b(n)x + theta(n)), where g is a periodic Lipschitz real function and a(n+1)/a(n) -> 0, b(n+1)/b(n) -> infinity as n -> infinity. Moreover, for any 1 <= H <= B <= 2 we provide examples of such functions with dim(H)(graph f) = (dim) under bar (B) (graph f) = H, (dim) over bar (B)(graph f) = B.