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Problem 1191 — Erdős Problems

Problem 1191

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 $c>0$ ?

Source: erdosproblems.com/1191

Additive combinatorics
Sidon sets

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No claim settles this problem.

Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
HaRo66: Halberstam, H. and Roth, K. F., Sequences. Vol. I. (1966), xx+291.

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Status: Open
Topics: Additive combinatorics
Sidon sets
In the wiki: Wiki page