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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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