Finite generation of split F-regular monoid algebras

Let S be a submonoid of a free Abelian group of finite rank. We show that if k is a field of prime characteristic such that the monoid k-algebra $k[S]$ is split F-regular, then $k[S]$ is a finitely generated k-algebra, or equivalently, that S is a finitely generated monoid. Split F-regular rings are possibly non-Noetherian or non-F-finite rings that satisfy the defining property of strongly F-regular rings from the theories of tight closure and F-singularities. Our finite generation result provides evidence in favor of the conjecture that split F-regular rings in function fields over k have to be Noetherian. The key tool is Diophatine approximation from convex geometry.