On a Problem of Sidon in Additive Number Theory and on Some Related Problems Addendum

In a note in this Journal [16 (1941), 212-215], Turan and I proved, among other results, the following : Let a l < a2 < . . . < a, < n be a sequence of positive integers such that the sums aj+a; are all different . Then x < n'1 +0(n1 ) . On the other hand, there exist such sequences with x >n1(2---e), for any e >0 . Recently I noticed that J . Singer, in his paper "A theorem in finite projective geometry and some applications to number theory" [Trans . Amer . Math . Soc., 43 (1938), 377-385], proves, among other results, that, if m is a power of a prime, then there exist m-{-1 numbers aI < a2 < . . . < am+, < m2 -f M+l such that the differences a i -a, are congruent, mod (m2 +M+1), to the integers 1, 2, . . ., m2+M. Clearly the sums a i+a; are all different, and since the quotient of two successive primes tends to 1, Singer's construction gives, for any large n, a set with x. > n2 (1-e), for any e > 0 . Singer's method is quite different from ours . His result shows that the above upper bound for x is best possible, except perhaps for the error term O(nl) .