扩张多项式与离散化Elekes-Rónyai定理
Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem
摘要：
我们的特点时，二元实解析函数是“维扩展”时，应用到笛卡尔积。如果$P$是一个二元实解析函数，它不是局部的$P(x,y) = h(a(x) + b(y))$形式，那么只要$A$和$B$是$\mathbb{R}$的具有豪斯多夫维数$0<α<1$的Borel子集，我们就有，对于独立于$P$的某个$ε(α)>0$，$P(A,B)$的豪斯多夫维数至少为$α+ ε$。结果是尖锐的，在这个意义上，没有这种形式的估计可以保持，如果$P(x,y) = h(a(x) + b(y))$。我们还证明了一个更技术的单尺度版本的结果，这是一个类似的Elekes-Rónyai定理的设置的Katz-陶离散环猜想。作为一个应用，我们证明了离散化的非集中集在三个不同的解析投影函数下不可能有小的非线性投影，只要相应的3-网具有非零的Blaschke曲率.
英文摘要：
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.
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