LAUSR

We introduce the concept of mixed sumset and estimate the lower bound for its cardinality by means of the polynomial method. The result generalizes the well known Cauchy–Davenport Theorem and a theorem of Alon, Nathanson and Ruzsa regarding the lower bound for the cardinality of restricted sumsets of distinct sets in a field $$\mathbb{F}$$ . As a consequence of this result, we also obtain a new proof for the estimation of lower bound for the cardinality of generalized h-fold sumset modulo a prime.