Assouad and Lower Dimensions of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies

We study the Assouad and lower dimensions of graphs of Weierstrass-type functions of the form $$f(x)=\sum _{n=1}^{\mathrm{\infty }}{a}_n\phi (b_{n}x+{\theta }_n),x\in \mathbb{R},$$ where $\phi$ is a nonconstant $C^2$ function of period one, $a_n,b_n>0$, $({\theta }_n)$ is an arbitrary sequence of real numbers, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n<\mathrm{\infty }$, and $b_{n+1}\mathrm{/}{b}_n\to \mathrm{\infty }$. We give quantitative conditions under which the graph over every nondegenerate compact interval has Assouad dimension two or lower dimension one. In particular, when $a_n=b_n^{-\alpha }$, we obtain explicit frequency-gap conditions under which the Assouad spectrum of the graph is equal to $2$ for $\alpha \le \vartheta <1$. When the logarithmic frequency ratios converge, we also determine the spectrum on a nonempty interval below $\alpha$. Together with Barański’s formulas for the Hausdorff and box dimensions, these results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.