The existing results are surveyed and several new results are proved for th
e cardinality of the restricted doubling 2 boolean AND A = {a' + a":a', a"
epsilon A, a' not equal a"}, where A subset of or equal to G is a subset of
the set of elements of an (additively written) group G. In particular, kno
wn estimates for G = Z and G = Z/pZ are improved and a first-of-a-kind gene
ral estimate valid for arbitrary G is given.