On the dimension of the graph of the classical Weierstrass function

This paper examines dimension of the graph of the famous Weierstrass non-differentiable functionWλ,b(x)=∑n=0∞λncos⁡(2πbnx) for an integer b≥2b≥2 and 1/b<λ<11/b<λ<1. We prove that for every b   there exists (explicitly given) λb∈(1/b,1)λb∈(1/b,1) such that the Hausdorff dimension of the graph of Wλ,bWλ,b is equal to D=2+log⁡λlog⁡b for every λ∈(λb,1)λ∈(λb,1). We also show that the dimension is equal to D for almost every λ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the functionf(x)=∑n=0∞λnϕ(bnx) for an integer b≥2b≥2 and 1/b<λ<11/b<λ<1 is equal to D   for a typical ZZ-periodic C3C3 function ϕ.