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 $0 <lambda<1$, $N_b,in,N$ and $lambda,N_b >1$, using a sequence a graphs that approximate the studied one.