erdos_340

Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the order of growth of $A$? Is it true that $|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$?

theorem erdos_340 (ε : ℝ) (hε : ε > 0) :
    (fun n : ℕ ↦ √n / n ^ ε) =O[atTop]
      fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ)

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