Erdős problem 868 · problems.science

Let $A$ be an additive basis of order $2$ , let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$ . If $f(n) > \epsilon \log n$ for large $n$ and an arbitrary fixed $\epsilon > 0$ , then must $A$ contain a minimal additive basis of order $2$ ?

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FormalConjectures/ErdosProblems/

868.lean

Retained formal statement1 of 4

Let $A$ be an additive basis of order $2$ , let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$ . If $f(n) \to \infty$ as $n \to \infty$ , then must $A$ contain a minimal additive basis of order $2$ ?

Larsen and Larsen [LaLa26] answered this in the negative.

FormalConjectures/ErdosProblems/868.leanErdos868.erdos_868.parts.i5 linesExact file

1False ↔2  ∀ (A : Set ℕ),3    A.IsAsymptoticAddBasisOfOrder 2 →4      Filter.Tendsto (fun n => Erdos868.ncard_add_repr A 2 n) Filter.atTop Filter.atTop →5        ∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2

SolvedStatement only, no proof

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