B2[g]SEQUENCES WHOSE TERMS ARE SQUARES

INTRODUCTION Sixty years ago Sidon [7] asked, in the course of some investigations of Fourier series, for a sequence a1 < a2... for which the sums ai + aj are all distinct and for which ak tends to infinity as slowly as possible. Sidon called these sequences, B2 sequences. The greedly algorithm gives ak ¿ k and this was the best result until Atjai, Komlos and Szemeredi [1] found a B2 sequence satisfying ak = o(k). However, this result is far from the main conjecture about B2 sequences. Conjecture. Corresponding to every 2 > 0, there exists a B2 sequence A such that aj ¿ j. In general we say that a sequence A is a B2[g] sequence if rn(A) ≤ g for all integer n, where rn(A) is the number of representations of n in the form n = a + b, a ≤ b (a, b ∈ A). In 1960, P.Erdös and A.Renyi [5], using probabilistic methods, proved the following first steps towards the conjecture.