The high-dimensional Weierstrass functions

For a real analytic periodic function φ:Rᵈ, an integer b ≥ 2 and λ∈(1/b,1), we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function W(x)=∑ₙ₌₀∞λⁿφ(bⁿx) are both equal to minlog_λ⁻¹b,\,1+(\,d-q\,)(1+logbλ)\, where q = q(φ, b, λ) denotes the maximum dimension of all linear spaces V < Rᵈ such that the projection πV W is Lipschitz.