We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(\delta,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$

中文翻译：

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关于离散和积问题

我们给出了实数集离散化环定理的新证明。作为一个特例，我们证明如果 $A\subset\mathbb{R}$ 是 Katz 和 Tao 意义上的 $(\delta,1/2)_1$-set，则 $A+A$ 或$AA$ 必须至少有 $|A|^{1-\frac{1}{68}}$