Bypassing dynamical systems: a simple way to get the box-counting dimension of the graph of the Weierstrass function

Abstract

In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by \[{\mathcal W}(x)= \sum_{n=0}^{+\infty} \lambda^n\,\cos \left ( 2\, \pi\,N_b^n\,x \right),\] where $\lambda$ and $N_b$ are two real numbers such that $0 1$, using a sequence a graphs that approximate the studied one.

Keywords

• Classical Weierstrass function
• box dimension