A Dichotomy for the Weierstrass-type functions

For a real analytic periodic function $$\phi :\mathbb {R}\rightarrow \mathbb {R}$$ , an integer $$b\ge 2$$ and $$\lambda \in (1/b,1)$$ , we prove the following dichotomy for the Weierstrass-type function $$W(x)=\sum \nolimits _{n\ge 0}{{\lambda }^n\phi (b^nx)}$$ : Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to $$2+\log _b\lambda $$ . Furthermore, given b and $$\phi $$ , the former alternative only happens for finitely many $$\lambda $$ unless $$\phi $$ is constant.