Erdős Problem 1146

References:

• erdosproblems.com/1146
• [Ru99] Ruzsa, I., Erdős and the Integers. Journal of Number Theory (1999), 115-163.

def Erdos1146.IsEssentialComponent (A : Set ℕ) : Prop

We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A \oplus B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density. Here, the sumset is the appropriate one for Schnirelmann density, $A \oplus B = \{a+b \mid a \in A \cup \{0\}, b \in B \cup \{0\}\}$ (i.e. $(A \cup \{0\}) + (B \cup \{0\})$). This avoids the trivial case where the sumset misses $1$ simply because neither $A$ nor $B$ contains $0$.

Equations

• Erdos1146.IsEssentialComponent A = ∀ (B : Set ℕ), have b := schnirelmannDensity B; 0 < b → b < 1 → schnirelmannDensity (A ∪ {0} + (B ∪ {0})) > b

theorem Erdos1146.erdos_1146 : sorry ↔ IsEssentialComponent {k : ℕ | ∃ (m : ℕ) (n : ℕ), k = 2 ^ m * 3 ^ n}

Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?

In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form $2^m3^n$ and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."