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~\mbox{$0<\lambda <1$},~\mbox{$N_b\in \mathbb{N}$} and~$\lambda N_b>1$, using a sequence a graphs that approximate the studied one.

arXiv admin note: substantial text overlap with arXiv:1703.06839, arXiv:1703.03371