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Erdős #1191 · additive combinatorics / sidon sets

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

lim inf⁡x→∞∣A∩[1,x]∣x1/2(log⁡x)1/2=0?\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 AA such that

lim inf⁡…\liminf_…
MSC 05 · Combinatorics

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

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

lim inf⁡x→∞∣A∩[1,x]∣x1/2(log⁡x)1/2=0?\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 AA such that

lim inf⁡x→∞∣A∩[1,x]∣x1/2(log⁡x)c>0\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^c>0

for some c>0c>0?

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Background

Erdős proved (see for example [HaRo66]) that if AA is an infinite Sidon set then

lim inf⁡x→∞∣A∩[1,x]∣x1/2(log⁡x)1/2≤c\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}(\log x)^{1/2}\leq c

for some constant c>0c>0. In [Er80] he offered $1000 'for clearing up the problems' raised by this; he may have meant for finding the optimal function ff such that

lim inf⁡x→∞∣A∩[1,x]∣x1/2f(x)=0\liminf_{x\to \infty} \frac{\lvert A\cap [1,x]\rvert}{x^{1/2}}f(x)=0

for all infinite Sidon sets AA.

The second question is a stronger form of [39]. See also [729] for the behaviour of the lim sup⁡\limsup.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1191 link

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