AI 中文总结
该论文结合单位距离与和积构造的算术工具，证明存在常数c>0使任意大复数集A同时满足加法和乘法集大小上界及单位距离对下界。
AI 中文摘要
存在绝对常数$c>0$，使得对于每个$0<\varepsilon\leq1$，任意大的有限集$A\subset\mathbb{C}$满足 \\[ |A+A|\leq |A|^{1+\varepsilon},\qquad |AA|\leq |A|^{2-c\varepsilon},\qquad \nu_1(A)\geq |A|^{1+c\varepsilon}, \\] 其中$\nu_1(A)$计数无序单位距离对。我们将近期单位距离构造中的算术方向与近期和积构造中使用的乘法放大相结合。算术成分作为显式输入给出。
英文摘要
There is an absolute constant $c>0$ such that, for every $0<\varepsilon\leq1$, arbitrarily large finite sets $A\subset\mathbb{C}$ satisfy \[ |A+A|\leq |A|^{1+\varepsilon},\qquad |AA|\leq |A|^{2-c\varepsilon},\qquad ν_1(A)\geq |A|^{1+c\varepsilon}, \] where $ν_1(A)$ counts unordered unit-distance pairs. We combine the arithmetic directions from the recent unit-distance construction with the multiplicative enlargement used in the recent sum-product construction. The arithmetic ingredients are stated as explicit inputs.
