Erdős Problem #1191 — Logarithmic density scale of infinite Sidon sets

If $A⊆N$ is an infinite Sidon set and $A(x)=|A∩[1,x]|$, must $\liminf_{x→∞} A(x)√{\log x}/√x=0$? Conversely, is there an infinite Sidon set and a constant $c>0$ for which $\liminf_{x→∞} A(x)(\log x)^c/√x>0$?