A nontrivial uniform algebra Dirichlet on its maximal ideal space

It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 64-year-old question of John Wermer and a 16-year-old question of Garth Dales and Joel Feinstein. Our example is P(X) for a certain compact set X in complex Euclidean 2-space ($\mathbb{C}^2$). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modified form of another 64-year-old question of Wermer.
Comment: The proof of the main theorem has been very slightly simplified and some minor expository improvements have been made