Additive group theory—A progress report

is solvable for every r. One easily obtains this result by setting A = B = {x:x = a(p)}. We then have \A\ = \B\ = (p + l)/2 and (2) follows from (1). Applying the C.-D. theorem to the representation of residues by sums of fcth powers one may without loss of generality restrict k to divisors of (p — 1). The C.-D. theorem then gives the result that every residue is a sum of not more than k kth powers. A considerable improvement is possible if one excludes the value k = (p — l)/2. G. A. Vosper [30], [31], [21] refined the C.-D. theorem by completely characterizing those pairs A, B for which