We demonstrate how additive number theory can be used to produce new classes of inequalities in the theory of enumeration of lattice points in polytopes. More specifically, we use a classical result of Kneser to produce new inequalities between the coefficients of the Ehrhart $h^*$-vector of a lattice polytope. As an application, we deduce all possible `balanced' inequalities between the coefficients of the Ehrhart $h^*$-vector of a lattice polytope containing an interior lattice point, in dimension at most $6$.