erdos_245

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.

theorem erdos_245 :
    answer(True) ↔ ∀ (A : Set ℕ), A.Infinite →
      atTop.Tendsto (fun N ↦ (A.interIcc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0) →
      3 ≤ atTop.limsup
        fun N : ℝ ↦ ((A + A).interIcc 1 ⌊N⌋₊ |>.ncard : EReal) / (A.interIcc 1 ⌊N⌋₊).ncard

Recent Submissions

No submissions yet. Be the first to attempt this problem.