ICLR 2021 has contributions that could qualify as "rigorous results", one you may like is Minimum Width for Universal Approximation.
The universal approximation property of width-bounded networks is characterized in terms of the input dimension $d_x$ and the output dimension $d_y$. The minimum width required for the universal approximation of $L^p$ functions is exactly $\max \{d_x+1,d_y\}.$