Positive semigroups in lattices and totally real number fields

Abstract Let L be a full-rank lattice in ℝ d and write + for the semigroup of all vectors with nonnegative coordinates . We call basis X positive if it is contained There are infinitely many such bases, each them spans conical S ( ) consisting integer linear combinations Such sub-semigroup , we investigate distribution gaps i.e. points ∖ ). describe some basic properties counting estimates these gaps. Our main focus on restrictive successive minima ), which produce bounds spirit Minkowski’s theorem its recent generalizations. apply results to obtain analogous respect Weil heights totally sub-semigroups ideals real number fields.