The \(\alpha\)-Weierstrass function is defined as
\(W^{\alpha,b}_g(x) = \sum_{k=0}^{\infty} b^{-\alpha k} g(b^k x)\),
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. By this we answer a question raised by R. Anttila, B. Bárány, and
A. Käenmäki. 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
\(x \mapsto \big(W^{\alpha,b}_{g_0}(x), W^{\alpha,b}_{g_1}(x), \dots, W^{\alpha,b}_{g_{d-1}}(x)\big)\)
is \(\alpha\)-bi-Hölder. We also prove that such an embedding requires at
least \(1/\alpha\) coordinate functions.
This is a joint work with Antti Käenmäki and Balázs Maga.