An elementary proof for the dimension of the graph of the classical Weierstrass function

2014/06/13 by Keller, Gerhard
#37D20 #37D45 #37G35 #37H20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1406.3571

Abstract

Let W_λ,b(x)=∑_n=0^∞λⁿ g(bⁿ x) where b\geqslant2 is an integer and g(u)=cos(2πu) (classical Weierstrass function). Building on work by Ledrappier (1992), Baránsky, Bárány and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of W_λ,b equals 2+(logλ)/(log b) for all λ∈(λ_b,1) with a suitable λ_b<1. This reproduces results by Baránsky, Bárány and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate.

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