Overview
This research investigates the box dimension of graphs associated with generalized Riemann-type functions, defined as $G_\delta(x)=\sum_{n=1}^{\infty}g(n^{2}x)n^{-1-\delta}$. These functions incorporate 1-periodic real-valued continuous functions $g$ and a parameter $0 < \delta \le 1$. The primary objective was to establish a rational-level criterion that informs the lower bound of the lower box dimension for the graphs of these functions.
Research Context
The study specifically addresses aspects related to the fractal geometry of functions, focusing on the box dimension, a measure of complexity for sets. The generalized Riemann-type functions under consideration are defined by an infinite series involving a base function $g(n^2x)$ and a decay factor $n^{-1-\delta}$. A notable context for this work is its direct engagement with a previously posed problem, identified as \cite[problem 2]{Wu-Zhan2026}, to which the findings provide a negative answer.
Approach
The research methodology centered on deriving a rational-level non-vanishing criterion. This criterion is applied to the square-class chirp functional, denoted $S_d(a;q)$. The analysis involved:
- Establishing a lower bound for the lower box dimension, $\dim_B(\mathrm{graph}\,G_\delta)$, under specific conditions.
- Utilizing a mild decay condition on the Fourier coefficients of the function $g$.
- Requiring the non-vanishing of the square-class chirp functional $S_d(a;q)$ at a single rational point $a/q$.
- Employing a resolution theorem to ascertain that for any nonconstant real trigonometric polynomial $g$, the chirp functional $S(a;q)$ cannot vanish simultaneously at every rational number.
Findings
The research yielded several specific findings regarding the box dimension of the graphs of generalized Riemann-type functions:
- A rational-level non-vanishing criterion was established for the lower bound of the lower box dimension of $\mathrm{graph}\,G_\delta$.
- Under a mild decay condition of the Fourier coefficients of $g$ and the non-vanishing of the square-class chirp functional $S_d(a;q)$ at a single rational $a/q$, the lower bound was proven to be $\dim_B(\mathrm{graph}\,G_\delta)\ge\frac74-\frac\delta2$.
- A resolution theorem indicated that for any nonconstant real trigonometric polynomial $g$, the chirp functional $S(a;q)$ cannot vanish at every rational simultaneously.
- Consequently, for all nonconstant real trigonometric polynomials $g$ and $0 < \delta \le 1$, the exact box dimension was determined to be $\dim_B(\mathrm{graph}\,G_\delta)=\frac74-\frac\delta2$. This result provides a negative answer to \cite[problem 2]{Wu-Zhan2026}.
- The study presented two guiding examples which differentiate between structural vanishing and genuinely arithmetic vanishing. These examples are noted to be related to modular elliptic curves and are governed by the Prime Number Theorem.
Why This Matters
This work provides a definitive answer to an existing problem in the field of fractal geometry related to generalized Riemann-type functions. By establishing a precise criterion and demonstrating its implications for a class of functions, the research refines the understanding of the box dimension for these complex mathematical structures. The distinction between types of vanishing phenomena, linked to number theory concepts like modular elliptic curves and the Prime Number Theorem, points to underlying arithmetic properties influencing fractal dimensions.