Differentiability and dimension of some fractal Fourier series (Q1283431)

This paper studies some analytic properties of a family of analogs of Riemann's function; these functions are defined via trigonometric series having frequencies in the \(k\)th powers, they are of the form: \(\sum c_n \frac{e(n^k x)}{n^\alpha}\), with \(0<\varliminf c_n\leq \varlimsup c_n<\infty\). The authors estimate the fractal dimension of their graphs, they study their Lipschitz order (under some further hypothesis on the coefficients \(c_n\)), and they give a complete characterization of the differentiability at rational values, when \(c_n= 1\), for every \(n\) and \(\alpha= k\). The proofs use classical tools in number theory and are based in particular on Weyl's inequalities. The results are nicely illustrated by computer graphics.