A simpler proof for the dimension of the graph of the classical Weierstrass function

Let $W_{\\lambda,b}(x)=\\sum_{n=0}^{\\infty}\\lambda^{n}g(b^{n}x)$ where $b\\geq2$ is an integer and $g(u)=\\cos(2\\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (In Symbolic Dynamics and Its Applications (1992) 285–293), Barański, Bárány and Romanowska (Adv. Math. 265 (2014) 32–59) and Tsujii (Nonlinearity 14 (2001) 1011–1027), we provide an elementary proof that the Hausdorff dimension of $W_{\\lambda,b}$ equals $2+\\frac{\\log\\lambda }{\\log b}$ for all $\\lambda\\in(\\lambda_{b},1)$ with a suitable $\\lambda_{b}<1$. This reproduces results by Barański, Bárány and Romanowska (Adv. Math. 265 (2014) 32–59) without using the dimension theory for hyperbolic measures of Ledrappier and Young (Ann. of Math. (2) 122 (1985) 540–574; Comm. Math. Phys. 117 (1988) 529–548), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate.