广义黎曼型函数图像盒维数的有理水平判据
A Rational-Level Criterion on Box Dimension of the Graph of Generalized Riemann-Type Functions
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中文总结 AI 辅助
该研究针对广义黎曼型函数图像的盒维数，建立有理水平非零判据，得出特定条件下盒维数的精确值，否定了Wu-Zhan2026的问题2，并区分了两类零点。
中文摘要 AI 辅助
我们研究广义黎曼型函数$G_\delta(x)=\sum_{n=1}^{\infty}g(n^2x)n^{-1-\delta}$的图像盒维数，其中$g$是1-周期实值连续函数，$0<\delta\le1$。首先，我们建立了$G_\delta$图像下盒维数下界的有理水平非零判据，具体而言，在$g$的傅里叶系数满足温和衰减条件，且单有理点$a/q$处的平方类啁啾泛函$S_d(a;q)$非零的情况下，证明$\dim_B(\text{graph}\\,G_\delta)\ge\frac74-\frac\delta2$。随后的判定定理表明，对于任何非常数实三角多项式$g$，啁啾泛函$S(a;q)$不可能在所有有理点同时为零；因此，对所有此类$0<\delta\le1$的$g$，$\dim_B(\text{graph}\\,G_\delta)=\frac74-\frac\delta2$，这给出了对文献[Wu-Zhan2026]中问题2的否定回答。最后，两个引导性例子区分了与模椭圆曲线相关、由素数定理支配的结构零点和真正算术零点。
英文摘要
We consider the box dimension of the graphs of the generalized Riemann-type functions $G_δ(x)=\sum_{n=1}^{\infty}g(n^{2}x)n^{-1-δ}$ with 1-periodic real-valued continuous functions $g$ and $0<δ\le 1$. Firstly, we establish a rational-level non-vanishing criterion for the lower bound of lower box dimension of the graph of $G_δ$. More precisely, We prove that the lower bound $\dim_B(\mathrm{graph}\,G_δ)\ge\frac74-\frac\delta2$ 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, $\dim_B(\mathrm{graph}\,G_δ)=\frac74-\frac\delta2$ for all such $g$ with $0<δ\le1$ 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.
