The Bouligand dimensions of the graphs of a class of functions (Q2714881)

In this paper the author gives the calculating formulae of the upper and lower Bouligand dimensions of the graph of trigonometric series NEWLINE\[NEWLINEf(x)=\sum_{j=1}^\infty a_j \cos (\lambda_j x-\phi_j),\qquad (a_j\in {\mathbb{R}}, \;\lambda_j\in {\mathbb{R}}^{+}, \;\phi_j\in [0, 2\pi], \;x\in [0, 2\pi]).NEWLINE\]NEWLINE Let \(\Gamma =\{(x, f(x))\in {\mathbb{R}}^2: x\in [0, 2\pi]\} \) and \( r_n=|a_n|\), \(n\in {\mathbb{N}}\). If \(r_{n+1}\lambda_{n+1}\geq 2.5 r_{n}\lambda_{n}\) and \( r_n\geq 2.4 r_{n+1}, \) then the upper and lower Bouligand dimensions of the graph \(\Gamma\) are given, respectively, by NEWLINE\[NEWLINE\Delta (\Gamma)=2+\limsup_{n\to \infty}\frac{\log r_n}{\log \lambda _n},\quad \text{and}\quad \delta (\Gamma)=2+\liminf_{n\to \infty}\frac{\log r_{n+1}}{\log r_n \lambda _n-\log r_{n+1}}.NEWLINE\]