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

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.