Problem 329

Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can

\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}

be?

Status

Open

No claim settles this problem.

CiTr01: Cilleruelo, Javier and Trujillo, Carlos, Infinite $B_2[g]$ sequences. Israel J. Math. (2001), 263-267.
Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
ErTu41: Erdős, P. and Turán, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.
Ko96: Kolountzakis, Mihail N., On the additive complements of the primes and sets of similar growth. Acta Arith. (1996), 1-8. The site's commentary credits Kolountzakis under the key [Ko96] with an infinite $B_2[2]$ sequence whose $\limsup$ of $|A\cap\{1,\ldots,N\}|/N^{1/2}$ is $1$ ; the site's reference record resolves the key to this paper on additive complements of the primes (Acta Arith. 77), the entry it shares with Problem 32. The construction is Theorem 4 of M. N. Kolountzakis, The density of $B_h[g]$ sequences and the minimum of dense cosine sums, J. Number Theory 56 (1996), 4-11, DOI 10.1006/jnth.1996.0002, which [CiTr01] lists as its reference [4] (library card).
Kr61: Krückeberg, Fritz, $B\sb{2}$ -Folgen und verwandte Zahlenfolgen. J. Reine Angew. Math. (1961), 53-60.