论文标题
有限西顿集中元素的总和II
Sum of elements in finite Sidon sets II
论文作者
论文摘要
如果所有总和$ a+b ~~（a，b \ in s）$不同，则设置$ s \ subset \ {1,2，...，n \} $被称为sidon集。令$ s_n $为$ \ {1,2，...，n \} $中的Sidon集的最大基数。在以前的一篇文章中，作者证明了以下渐近公式$$ \ sum_ {a \ in s，〜| s | = s_n} a = \ frac {1} {2} {2} n^{3/2}+o（n^{n^{111/80+\ varepsilon}）在本说明中，我们给出上述公式的扩展。我们表明$$ \ sum_ {a \ in s，〜| s | = s_n} a^{\ ell} = \ frac {1} {\ ell+1} n^{\ ell+1/2}+o \ left（n^{n^{\ ell+61/160} \ right）此外，我们还考虑了涉及西顿集合的其他类型求和的渐近公式。这些证明是在更通用的环境中建立的，即，当$ t $接近$ n^{1/2} $时，我们获得了带有$ t $元素的Sidon集的渐近公式。
A set $S\subset\{1,2,...,n\}$ is called a Sidon set if all the sums $a+b~~(a,b\in S)$ are different. Let $S_n$ be the largest cardinality of the Sidon sets in $\{1,2,...,n\}$. In a former article, the author proved the following asymptotic formula $$\sum_{a\in S,~|S|=S_n}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$$ where $\varepsilon>0$ is an arbitrary small constant. In this note, we give an extension of the above formula. We show that $$\sum_{a\in S,~|S|=S_n}a^{\ell}=\frac{1}{\ell+1}n^{\ell+1/2}+O\left(n^{\ell+61/160}\right)$$ for any positive integers $\ell$. Besides, we also consider the asymptotic formulae of other type summations involving Sidon sets. The proofs are established in a more general setting, namely we obtain the asymptotic formulae of the Sidon sets with $t$ elements when $t$ is near the magnitude $n^{1/2}$.
