Problem 1191

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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 $c>0$?

Status. Open.

Source. erdosproblems.com/1191, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1191, https://www.erdosproblems.com/1191.

References.

• [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.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

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