Problem 39

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that

\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}

for all $\epsilon>0$ ?

Status

Open

No claim settles this problem.

AKS81b: Ajtai, Miklós and Komlós, János and Szemerédi, Endre, A dense infinite Sidon sequence. European J. Combin. (1981), 1-11.
Er73: Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
Er77c: Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section C9 "Packing sums of pairs", printed p. 176: the infinite case, with the Erdős--Turán bound $\limsup a_k/k^2=\infty$ and the Ajtai--Komlós--Szemerédi sequence with $a_k<ck^3/\ln k$ . Library home: Guy 2004.
Ru98: Ruzsa, Imre Z., An infinite Sidon sequence. J. Number Theory (1998), 63-71.