Lately, there has been a lot of interest in the weighted
Birkhoff averages of form
\[A_{w,\phi}(x)=\lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} w_i
\phi(T^ix),\]
where $\phi$ is a potential and $(w_i)$ is a fixed sequence of real
numbers (weights).
In this talk I'll present results (joint with Ruxi Shi and Balazs Barany)
about the multifractal spectrum of the weighted Birkhoff averages.