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On the Hausdorff dimension of some graphs. (English) Zbl 0603.28003

Consider the functions

W b (x)= n=- b -αn [Φ(b n x+θ n )-Φ(θ n )],

where b>1, 0<α<1, each θ n is an arbitrary number, and Φ has period one. We show that there is a constant C>0 such that if b is large enough, then the Hausdorff dimension of the graph of W b is bounded below by 2-α-(C/lnb). We also show that if a function f is convex Lipschitz of order α, then the graph of f has σ- finite measure with respect to Hausdorff’s measure in dimension 2-α. The convex Lipschitz functions of order α include Zygmund’s class Λ α . Our analysis shows that the graph of the classical van der Waerden-Takagi nowhere differentiable function has σ-finite measure with respect to h(t)=t/ln(1/t).


MSC:
28A75Length, area, volume, other geometric measure theory
42A32Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives