erdos_12.variants.example

An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ and such that \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N > 0\] is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.

theorem erdos_12.variants.example (A : Set ℕ)
    (hA : A = {p ^ 2 | (p : ℕ) (_ : p.Prime) (_ : p ≡ 3 [MOD 4])}) :
    IsGood A ∧ 0 < atTop.liminf (fun (N : ℕ) ↦ (A.interIcc 1 N).ncard * (N : ℝ).log / √N)

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