论文标题
扩展维度的多项式和离散的Elekes-rónyai定理
Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem
论文作者
论文摘要
我们表征何时将双变量实际分析函数应用于笛卡尔产品时“维度扩大”。如果$ p $是$ p（x，y）= h（a（x） + b（y））$的本地的双变量真实分析功能，那么每当$ \ mathbb {r} $ a $ a $ a $ a $ a $ a和$ b $的子集与hausdorff dimersion $ 0 <α<α<1 $ p（对于某些$ε（α）> 0 $，独立于$ p $。结果是尖锐的，从某种意义上说，如果$ p（x，y）= h（a（x） + b（y））$，则无法估算此形式。我们还证明了此结果的更具技术性的单尺度版本，这是elekes-rónyai定理在Katz-Tao离散戒指猜想的情况下的类似物。作为应用程序，我们表明，在三个不同的分析投影函数下，离散的非浓缩集不能具有小的非线性投影，前提是相应的3-WEB具有非变化的Blaschke曲率。
We characterize when bivariate real analytic functions are "dimension expanding" when applied to a Cartesian product. If $P$ is a bivariate real analytic function that is not locally of the form $P(x,y) = h(a(x) + b(y))$, then whenever $A$ and $B$ are Borel subsets of $\mathbb{R}$ with Hausdorff dimension $0<α<1$, we have that $P(A,B)$ has Hausdorff dimension at least $α+ ε$ for some $ε(α)>0$ that is independent of $P$. The result is sharp, in the sense that no estimate of this form can hold if $P(x,y) = h(a(x) + b(y))$. We also prove a more technical single-scale version of this result, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture. As an application, we show that a discretized non-concentrated set cannot have small nonlinear projection under three distinct analytic projection functions, provided that the corresponding 3-web has non-vanishing Blaschke curvature.
