k-Fold Sidon Sets

Let k > 1 be an integer. A set A ⊂ Z is a k-fold Sidon set if A has only trivial solutions to each equation of the form c1x1 + c2x2 + c3x3 + c4x4 = 0 where 0 6 |ci| 6 k, and c1 + c2 + c3 + c4 = 0. We prove that for any integer k > 1, a k-fold Sidon set A ⊂ [N ] has at most (N/k)1/2 + O((Nk)1/4) elements. Indeed we prove that given any k positive integers c1 < · · · < ck, any set A ⊂ [N ] that contains only trivial solutions to ci(x1 − x2) = cj(x3 − x4) for each 1 6 i 6 j 6 k, has at most (N/k)+O((ckN/k) 1/4) elements. On the other hand, for any k > 2 we can exhibit k positive integers c1, . . . , ck and a set A ⊂ [N ] with |A| > ( 1 k +o(1))N 1/2, such that A has only trivial solutions to ci(x1 − x2) = cj(x3 − x4) for each 1 6 i 6 j 6 k.