论文标题
不对称估计和总产量问题
Asymmetric estimates and the sum-product problems
论文作者
论文摘要
我们显示了两个不对称估计值，一个是在共线三元的数量上，另一个是在解决方案的估计数中$（a_1+a_2）（a_1^{\ prime \ prime \ prime}+a_2^{\ prime \ prime \ prime \ prime \ prime}）=（a_1^\ prime+a_2^\ prime）（a_1^{a_1^{\ prime \ prime \ prime}+a_2^+a_2^^{\ prime \ prime \ prime \ prime}）$。作为应用程序，我们改善了差异/分层估计和Balog-wooley分解的结果：对于任何有限的子集$ a $ a $ a $ \ shatbb {r} $，\ [\ max \ [\ max \ {| a-a-a |，| aa | aa | \ \ \ \ \ \ \ \ \ \ \ gtrsim \ max \ {| a-a |，| a/a | \} \ gtrsim | a |^{1+15/49}。此外，还有sets $ b，c $ with $ a = b \ sqcup c $，以便\ [\ max \ {e^+（b），\，e^\ times（c）\} \ lyseSim | a | a |^{3-3/11}。 \]
We show two asymmetric estimates, one on the number of collinear triples and the other on that of solutions to $(a_1+a_2)(a_1^{\prime\prime\prime}+a_2^{\prime\prime\prime})=(a_1^\prime+a_2^\prime)(a_1^{\prime\prime}+a_2^{\prime\prime})$. As applications, we improve results on difference-product/division estimates and on Balog-Wooley decomposition: For any finite subset $A$ of $\mathbb{R}$, \[ \max\{|A-A|,|AA|\} \gtrsim |A|^{1+105/347},\quad \max\{|A-A|,|A/A|\} \gtrsim |A|^{1+15/49}. \] Moreover, there are sets $B,C$ with $A=B\sqcup C$ such that \[ \max\{E^+(B),\, E^\times (C)\} \lesssim |A|^{3-3/11}. \]
