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on Erdős #1191 · additive combinatorics / sidon sets

The statement

Let $A\subset\mathbb{N}$ be an infinite Sidon set. Is it true that

\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^{1/2}=0?

Does there exist an infinite Sidon set $A$ such that

\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^c>0

for some $c>0$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.