Combining two elementary proofs we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.

中文翻译：

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西顿集大小的上限

结合两个基本证明，我们在经典组合数论问题中将上下界之间的距离减少了0.2 \％$。我们表明，对于足够大的$ n $，西顿集合$ \ {1，2，\ ldots，n \} $的最大大小最多为$ \ sqrt {n} + 0.998n ^ {1/4} $ 。