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An “Average” Erdős–Turán conjecture
Right, so the Erdos-Turan conjecture for additive bases (of order 2) says, with the usual notations, that sup r_B (n) = \infty. Let's look instead at the average number of representations, i.e.: the function
F(N) = 1/N * \sum_{n=1}^{N} r_B (n)
This seems entirely natural to me, indeed more natural than looking at (lim)sup r_B (n), if the idea is to demonstrate that a basis must have some "thickness". There are examples known where sup F(N) < \infty, indeed any so-called "thin" basis has this property. Recall that a basis is "thin" if there exists c > 0 such that the k:th element of the basis is at least c * k^2. Now an obvious question to ask is whether limsup F(N) can equal one ? (it must be at least one if B is a basis). I have searched the literature in vain for an answer. In particular, I have not found any result on thin bases which directly translates into an answer to this question, though that may because I am too stupid to see some connection. Any comments most welcome.
