Dirichlet regularity in arbitrary o‐minimal structures on the field ℝ up to dimension 4

In this article we show that the set of Dirichlet regular boundary points of a bounded domain of dimension up to 4, definable in an arbitrary o‐minimal structure on the field ?, is definable in the same structure. Moreover we give estimates for the dimension of the set of non‐regular boundary points, depending on whether the structure is polynomially bounded or not. This paper extends the results from the author's Ph.D. thesis 6, 7] where the problem was solved for polynomially bounded o‐minimal structures expanding the real field. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)