Erdős Problem #125

Let \(A\) be the set of integers which have only the digits \(0, 1\) when written base 3, -/ local notation "A" => { x : ℕ | (digits 3 x).toFinset ⊆ {0, 1} } /-- and \(B\) be the set of integers which have only the digits \(0, 1\) when written base 4. -/ local notation "B" => { x : ℕ | (digits 4 x).toFinset ⊆ {0, 1} } /- There are four possibilities for the density of \(A+B\): 1. \(A+B\) has zero upper and lower density (and hence also zero density). 2. \(A+B\) has zero lower density, but positive upper density (and hence no density). 3. \(A+B\) has positive upper and lower density that are equal (and hence positive density). 4. \(A+B\) has positive upper and lower density that are unequal (and hence no density). -/ /-- Case 3: Does \(A + B\) have positive upper and lower density that are equal? This is the literal interpretation of "positive density" which was falsified.