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

摘要

Let W-lambda,W-b(x) = Sigma(infinity)(n=0) lambda(n) g (b(n) x) where b >= 2 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), Baraliski, Barany 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,W-b equals 2+ log lambda/log b, for all lambda is an element of (lambda(b), 1) with a suitable lambda(b) < 1. This reproduces results by Baraiiski, Barany 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.