More precisely, are there "interesting" theorems of Presburger arithmetic, other than the following four well-known "interesting" ones?
The commutativity of addition.
The theorem stating there are no two consecutive even numbers.
The theorem stating that every natural number, either is even or is followed by an even successor (along with analogous theorems about a sum of more than two identical natural numbers, e.g., the theorem stating that every natural number not being any sum of three identical natural numbers - is followed by a successor that either is such a sum or is followed by a successor which is such a sum).
For our purposes, for an "additive" theorem to be "interesting" enough: 1) it must have an intuitive meaning [like that of the theorems I've mentioned as examples]; 2) its proof should need to rely on induction; 3) It should not be an intuitive generalization of any axiom [e.g. the axiom x+(y+1)=(x+y)+1, which can be viewed as a special case wherein z=1, intuitively generalized - for all z - as the theorem of associativity of addition: x+(y+z)=(x+y)+z].