Erdős problem 329 · problems.science

Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2} be?

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6 retained statements2415f78e850a

FormalConjectures/ErdosProblems/

329.lean

Retained formal statement2 of 6

The converse: if the maximum density is 1, then any finite Sidon set can be embedded in a perfect difference set modulo $n > 0$ .

Since the consequent is false (due to the counterexamples in [Ha47] and [AlMi25]), this implication is logically equivalent to the statement that the maximum upper density of Sidon sets is NOT 1. Because the maximum upper density problem is still open, the truth value of this implication is also an open research problem.

FormalConjectures/ErdosProblems/329.leanErdos329.erdos_329.variants.converse_implication2 linesExact file

1sSup {x | ∃ A, ∃ (_ : IsSidon A), Erdos329.sidonUpperDensity A = x} = 1 →2  ∀ (A : Finset ℕ), IsSidon ↑A → ∃ D n, ∃ (_ : n > 0), ↑A ⊆ D ∧ IsPerfectDifferenceSet D n

OpenStatement only, no proof

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