FormalConjectures.ErdosProblems.«31»

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import FormalConjecturesUtil

Erdős Problem 31

References:

erdosproblems.com/31

[Er56] Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137.

[Er59] Erdős, P., Über einige Probleme der additiven Zahlentheorie. Sammelband zu Ehren des 250. Geburtstages Leonhard Eulers (1959), 116-119.

[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.

[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.

[Lo54] Lorentz, G. G., On a problem of additive number theory. Proc. Amer. Math. Soc. (1954), 838-841.


namespace Erdos31


open Filter

open scoped Pointwise

Given any infinite set $A\subset \mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ contains all except finitely many integers.

Conjectured by Erdős and Straus. Proved by Lorentz [Lo54].

@[category research solved, AMS 11,
  formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos31.lean"]
theorem declaration uses 'sorry'erdos_31 : ∀ A : Set ℕ, A.Infinite →
    ∃ B : Set ℕ, B.HasDensity 0 ∧ ∀ᶠ n in atTop, n ∈ A + B := by⊢ ∀ (A : Set ℕ), A.Infinite → ∃ B, B.HasDensity 0 ∧ ∀ᶠ (n : ℕ) in atTop, n ∈ A + B
  sorryAll goals completed! 🐙


end Erdos31