Some generalizations of the Cauchy–Davenport theorem

We investigate two possible generalizations of the Cauchy–Davenport inequality $|A+B|\geq\min(p,|A|+|B|-1)$ for nonempty sets $A,B$ of residues modulo a prime number $p$. The first one deals with another way of measuring the size of a set of points in an affine space (rather than just taking the cardinality), namely, with algebraic complexity. The second one concentrates on the multiplicative group of a field.