$L^1$ norm for product of cosines

Question

Let $k$ be an integer and consider the function $$ f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t). $$ I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to mind is using Hölder's inequality: $$ \int_{0}^{1} |f(t)| dt \le \prod_{i=1}^{k} \bigg(\int_{0}^{1}|\cos(3^{i-1}\pi t)|^k dt \bigg)^{\frac{1}{k}} $$ Now, using the periodicity of $|\cos(3^{i-1}\pi t)|$, note that $$ \int_{0}^{1}|\cos(3^{i-1}\pi t)|^k dt =\int_{0}^{1}|\cos(\pi t)|^k dt, $$ so that $$ \int_{0}^{1} |f(t)|dt \le \int_{0}^{1}|\cos(\pi t)|^k dt, $$ and I'm sure there should be some results for the moments of cosine. However, computationally this doesn't seem like a good bound. Is there a better way to do this? The reason I'm interested in this function is because it appears in the context of the Fourier transform of a certain set of integers.