Dense Infinite $B_h$ Sequences

Abstract

For $h=3$ and $h=4$ we prove the existence of infinite $B_h$ sequences $\mathcal{B}$ with counting function

$$\mathcal{B}(x)= x^{\sqrt{(h-1)^2+1}-(h-1) + o(1)}.$$

This result extends a construction of I. Ruzsa for $B_2$ sequences.