On box dimension of the graphs of the generalized Riemann-type functions

Abstract

We investigate the box dimension of the graphs of a class of continuous periodic functions $G_{\delta }(x)={\sum }_{n=1}^{\infty }g(n^{2}x)n^{-1-\delta }$ with 1-periodic Lipschitz functions $g$ and $0<\delta \le 1$, which generalizes the result of the classical Riemann function corresponding to $g(x)=\sin (2\pi x)$ and $\delta =1$. More precisely, we first prove that the lower box dimension of the graph of $G_{\delta }$ is no less than $\frac{7}{4}-\frac{\delta }{2}$ when the Fourier coefficients of $g$ satisfy an arithmetic non-vanishing condition related to the distribution of quadratic residues. This result is new and non-trivial even when $g$ has a finite Fourier expansion, highlighting the intrinsic arithmetic complexity of the series. Secondly, if $g^{\prime }$ is Lipschitz continuous on $\mathbb{R}$, we show that the upper box dimension does not exceed \(\frac74-\fracδ{2}\), which extends earlier work of Chamizo and Córdoba and reveals deep connection between the regularity of $g$ and the fractal dimension of the associated Riemann-type series. In the end, we give some illustrative examples and propose some further problems.