Sumsets in Vector Spaces over Finite Fields

AbstractWe determine explicitly the least possible size of the sumset of two subsetsA, B⊂(Z/pZ)Nwith fixed cardinalities, thereby generalizing both Cauchy–Davenport's theorem (caseN=1) and Yuzvinsky's theorem(casep=2). The solution involves a natural generalization of the well-known Hopf–Stiefel–Pfister function. The corresponding problem for more than two summands is also considered and solved. We then consider restricted sumsets, formed by taking sums of distinct elements only. We determine almost completely the least possible size of the restricted sumset of two subsets in (Z/pZ)Nwith fixed cardinalities. Our result generalizes the recent solution(s) of the Erdős–Heilbronn conjecture dealing with the restricted sumsets of two equal subsets inZ/pZ