Level sets of prevalent Weierstrass functions

The $\alpha$-Weierstrass function is defined as $W_g^{\alpha ,b}(x)={\sum }_{k=0}^{\mathrm{\infty }}{b}^{-\alpha k}g(b^{k}x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $\alpha$-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most $1-\alpha$, and the Hausdorff dimension of almost every level set equals $1-\alpha$ with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent $\alpha$-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension $d$, we construct Lipschitz functions $g_0,g_1,\dots ,g_{d-1}$ such that the mapping $x\mapsto (W_{g_0}^{\alpha ,b}(x),W_{g_1}^{\alpha ,b}(x),\dots ,W_{g_{d-1}}^{\alpha ,b}(x))$ is $\alpha$-bi-Hölder. We also prove that such an embedding requires at least $1\mathrm{/}\alpha$ coordinate functions.