On the dimension of the graph of the classical Weierstrass function (Q406289)

The authors study the Hausdorff dimension of the graph \(\Gamma(b,\lambda)\) of the famous Weierstrass function NEWLINE\[NEWLINE W_{b,\lambda}(x) =\sum_{n=0}^{\infty}\lambda^{n}\cos(2\pi b^{n}x).NEWLINE\]NEWLINE As known, for \(b>1\) and \(1/b<\lambda<1\) this function is continuous and nowhere differentiable on the real axis.NEWLINENEWLINE\textit{J. L. Kaplan} et al. [Ergodic Theory Dyn. Syst. 4, 261--281 (1984; Zbl 0558.58018)] proved that the box counting dimension of \(\Gamma(b,\lambda)\) is equal to NEWLINE\[NEWLINE D=2+\log_{b}{\lambda}.NEWLINE\]NEWLINE The authors of the present paper prove that for any integer \(b\geq 2\) there exist two constants \(\lambda_b\) and \(\tilde{\lambda}_b\) such that \(1/b < \tilde{\lambda}_b <\lambda_b <1\), and for all \(\lambda\in (\lambda_{b}, 1)\) and for almost all \(\lambda\in (\tilde{\lambda}_{b}, \lambda_{b}]\) the Hausdorff dimension of \(\Gamma(b,\lambda)\) also equals to \(D\). They obtain certain formulas for these constants and show that NEWLINE\[NEWLINE \lambda_{b} \to \frac{1}{\pi}, \quad \tilde{\lambda}_{b}\sqrt{b}\to \frac{1}{\sqrt{\pi}}NEWLINE\]NEWLINE for \(b\to\infty.\)NEWLINENEWLINEThe authors generalize some of these results for graphs of functions \(\sum\limits_{n=0}^{\infty}\lambda^{n}\phi(\lambda^{n}x)\), where \(\phi\) stands for sufficiently smooth 1-periodic function.