  In 1942 Mann solved a famous problem, the $\alpha+\beta$ conjecture, about
the lower bound of the Shnirel'man density of sums of sets of positive
integers. In 1945, Dyson generalized Mann's theorem and obtained a lower bound
for the Shnirel'man density of rank $r$ sumsets. His proof introduced a method
that evolved into the Dyson transform, an important tool in additive number
theory. This paper explains the background of Dyson's work, gives Dyson's proof
of his theorem, and includes several applications of the Dyson transform, such
as Kneser's inequality for sums of finite subsets of an arbitrary additive
abelian group.
