Erdős problem 245 · problems.science

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$ . Is it true that $\limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3?$

Sources

Browse retained paths and inspect the exact material available for this Problem.

2 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

245.lean

Retained formal statement1 of 2

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$ . Is it true that $\limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3?$

The answer is yes, proved by Freiman [Fr73].

[Fr73] Fre\u{\i}man, G. A., _Foundations of a structural theory of set addition_. (1973), vii+108.

FormalConjectures/ErdosProblems/245.leanErdos245.erdos_2455 linesExact file

1True ↔2  ∀ (A : Set ℕ),3    A.Infinite →4      Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard / N) Filter.atTop (nhds 0) →5        3 ≤ Filter.limsup (fun N => ↑((A + A) ∩ Set.Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard) Filter.atTop

SolvedStatement only, no proof

Sources

Search problems.science