A Rational-Level Criterion on Box Dimension of the Graph of Generalized Riemann-Type Functions

Abstract

We consider the box dimension of the graphs of the generalized Riemann-type functions $G_{\delta }(x)={\sum }_{n=1}^{\infty }g(n^{2}x)n^{-1-\delta }$ with 1-periodic real-valued continuous functions $g$ and $0<\delta \le 1$. Firstly, we establish a rational-level non-vanishing criterion for the lower bound of lower box dimension of the graph of $G_{\delta }$. More precisely, We prove that the lower bound ${\dim }_B(\mathrm{graph}{G}_{\delta })\ge \frac{7}{4}-\frac{\delta }{2}$ under a mild decay condition of the Fourier coefficients of $g$ and non-vanishing of the square-class chirp functional $S_d(a;q)$ at a single rational $a/q$. A resolution theorem then asserts that for any nonconstant real trigonometric polynomial $g$, the chirp functional $S(a;q)$ cannot vanish at every rational simultaneously; consequently, ${\dim }_B(\mathrm{graph}{G}_{\delta })=\frac{7}{4}-\frac{\delta }{2}$ for all such $g$ with $0<\delta \le 1$ which gives a negative answer to \cite[problem 2]{Wu-Zhan2026}. Finally, two guiding examples distinguish structural vanishing from genuinely arithmetic vanishing related to modular elliptic curve and governed by the Prime Number Theorem.