Erdős Problem #899

Let $A\subseteq \mathbb{N}$ be an infinite set such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that\[\limsup_{N\to \infty}\frac{\lvert (A-A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?\]

#899: [Er82e]

additive combinatorics

The answer is yes, proved by Ruzsa [Ru78].

See also [245] for the sum set analogue.

View the LaTeX source

External data from the database - you can help update this
Formalised statement? Yes

0 comments on this problem

Likes this problem	None
Interested in collaborating	None
Currently working on this problem	None
This problem looks difficult	None
This problem looks tractable	None

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #899, https://www.erdosproblems.com/899, accessed 2026-01-16