Let $k_i\in \mathbb{N}$ $(i⩾1)$ satisfy $2⩽k_1⩽k_2⩽\cdots$. Freĭman's theorem shows that when $j\in \mathbb{N}$, there exists $s=s(j)\in \mathbb{N}$ such that all large integers $n$ are represented in the form $n=x_1^{k_j}+x_2^{k_{j+1}}+\cdots +x_s^{k_{j+s-1}}$, with $x_i\in \mathbb{N}$, if and only if $\sum k_i^{-1}$ diverges. We make this theorem effective by showing that, for each fixed $j$, it suffices to impose the condition

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关于韦林问题：超越弗里曼定理

让$k_{我}\epsilon \mathbb{不是}$ $（我⩾1）$满足$2⩽k_1⩽k_2⩽\cdots$。弗里曼定理表明，当$j\epsilon \mathbb{不是}$， 那里存在$s=s（j）\epsilon \mathbb{不是}$使得所有大整数$\mathrm{不是}$表示为以下形式$\mathrm{不是}=X_1^{k_j}+X_2^{k_{j+1}}+\cdots +X_s^{k_{j+s-1}}$， 和$X_{我}\epsilon \mathbb{不是}$，当且仅当$\Sigma k_{我}^{-1}$发散。我们通过证明对于每个固定的$j$，只要施加条件即可