Carleman approximation on Riemann surfaces without critical points

Key Points

• Carleman approximation is feasible on semi-admissible subsets of open Riemann surfaces, expanding the context of non-critical holomorphic functions.
• Closed sets with empty interior can effectively approximate continuous functions using non-critical holomorphic functions, showcasing significant mathematical properties.
• An alternate approach allows uniform approximation by non-critical holomorphic functions on broader sets than previously established semi-admissible ones.
• This opens up new avenues for understanding the approximation of functions on complex surfaces, emphasizing the importance of non-critical behavior.