On the dimension of the graph of the classical Weierstrass function

الأصل · EN

This paper examines dimension of the graph of the famous Weierstrass non-differentiable function Wλ, b (x) = ∑ₙ₌₀∞λⁿ(2πbⁿ x) for an integer b ≥ 2 and 1/b < λ< 1. We prove that for every b there exists (explicitly given) λb ∈ (1/b, 1) such that the Hausdorff dimension of the graph of Wλ, b is equal to D = 2+λ/ b for every 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 function f (x) = ∑ₙ₌₀∞λⁿϕ(bⁿ x) for an integer b ≥ 2 and 1/b < λ< 1 is equal to D for a typical Z-periodic C³ function ϕ.