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On the complexification of the Weierstrass non-differentiable function. (English) Zbl 1038.28005
Consider the well-known Weierstrass nowhere differentiable functions \(X_{a,b}t=\sum_{n=0}^{\infty} \cos(b^nt)\) and \(Y_{a,b}t=\sum_{n=0}^{\infty} \sin(b^nt)\). It is known that the problem of computing the Hausdorff dimension of the graphs of these functions is still unsolved. Here it is shown that the set \((X_{a,b},Y_{a,b})([ 2,\pi ])\) has a non-empty interior in \(\mathbb{R}^2\), provided \(b \in \mathbb{N}\), \(b \geq 2\) and \(a<1\) is suffficiently close to 1. Also, it is shown that the Hausdorff dimension of graph\(X_{a,b}\) and graph\(Y_{a,b}\) is at least \(1+\alpha=-\frac{\log a}{\log b}\).

MSC:
28A80 Fractals
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
28A78 Hausdorff and packing measures
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