OPENNOTORIOUSnumber theorySidon setsadditive combinatorics
Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$, $h(N) = N^{1/2}+O_\epsilon(N^\epsilon)$?
Notes: Erdos and Turan proved h(N) <= N^{1/2} + O(N^{1/4}). Best bound: h(N) <= N^{1/2} + 0.98183*N^{1/4} + O(1) by Carter, Hunter, O'Bryant. Singer showed h(N) >= (1-o(1))N^{1/2}.
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