On $B_{2k}$-sequences

Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a $B_r$-sequence A satisfies $lim inf_{n→ ∞} (A(n)/(n^{1/r}))=0$. The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof of Erdős' hypothesis for all even r=2k which we developped independently of Jia's version.