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 $G_δ(x)=\sum_{n=1}^{\infty}g(n^{2}x)n^{-1-δ}$ with 1-periodic Lipschitz functions $g$ and $0<δ\le 1$, which generalizes the result of the classical Riemann function corresponding to $g(x)=\sin(2πx)$ and $δ=1$. More precisely, we first prove that the lower box dimension of the graph of $G_δ$ is no less than $\frac74-\fracδ{2}$ 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 $g'$ 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.