On the Hausdorff dimension of some graphs

Consider the functions Wb(x)= b-cn[1(bnX + On)--1(0n)] n=-oo where b > 1, 0 0 such that if b is large enough, then the Hausdorff dimension of the graph of Wb is bounded below by 2a (C/ ln b). We also show that if a function f is convex Lipschitz of order a, then the graph of f has a-finite measure with respect to Hausdorff's measure in dimension 2 a. The convex Lipschitz functions of order a include Zygmund's class A,. Our analysis shows that the graph of the classical van der WaerdenTagaki nowhere differentiable function has a-finite measure with respect to h(t)-=t/ In(1/t). We consider the Hausdorff dimension of the graphs of various continuous functions. We introduce a new geometric property of a function: convex Lipschitz of some order, and obtain an upper bound on the dimension of a graph with this property. In particular, our analysis includes functions of the form

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