We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin. (3) The parallelizability of planar d-webs. (4) The Elekes-Rónyai theorem on expanding polynomials.

Given a Borel set A in the plane, we study the set of exceptional vantage points, for which the pinned distance Δ_p(A) has small dimension, that is, close to (dimA)/2. We show that if this set has positive dimension, then it must have very special structure. This result follows from a more general single-scale nonlinear projection theorem, which says that if ϕ₁, ϕ₂, ϕ₃ are three smooth functions whose associated 3-web has non-vanishing Blaschke curvature, and if A is a (δ,α)₂-set in the sense of Katz and Tao, then at least one of the images ϕ_i(A) must have measure much larger than |A|^1/2, where |A| stands for the measure of A. We prove analogous results for d smooth functions ϕ₁,…,ϕ_d, whose associated d-web is not parallelizable.

We use similar tools to characterize when bivariate real analytic functions are “dimension expanding” when applied to a Cartesian product: if P is a bivariate real analytic function, then P is either locally of the form h(a(x)+b(y)), or P(A,B) has dimension at least α+c whenever A and B are Borel sets with Hausdorff dimension α. Again, this follows from a single-scale estimate, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture.

我们考虑四个相关问题。 (1) 获得固定 Falconer 距离问题的一组特殊有利点的维度估计。 (2) 非线性投影定理，本着考夫曼、布尔干和什默金的精神。 (3)平面d -webs的并行性。 (4) 多项式展开式的 Elekes-Rónyai 定理。

给定平面上的Borel 集A ，我们研究一组特殊的有利点，其中固定距离 Δ _p ( A ) 具有较小的尺寸，即接近 (dim A )/2。我们证明，如果这个集合具有正维数，那么它一定具有非常特殊的结构。这个结果来自一个更一般的单尺度非线性投影定理，该定理说，如果phi ₁、phi ₂、phi ₃是三个平滑函数，其相关的 3-web 具有非零 Blaschke 曲率，并且如果A是 ( δ , α ) ₂ -Katz 和 Tai 意义上的集合，则至少一幅图像phi _i ( A ) 的度量必须远大于 |一个| ^1/2，其中 |一个|代表A的度量。我们证明了d平滑函数ψ ₁ ,…, ψ _d的类似结果，其相关的d -web 不可并行化。

我们使用类似的工具来表征二元实数解析函数在应用于笛卡尔积时何时“维数扩展”：如果P是二元实数解析函数，则P的局部形式为h ( a ( x )+ b ( y ))，或者只要A和B是 Hausdorff 维数为α的 Borel 集， P ( A , B ) 的维数至少为α + c。同样，这是根据单尺度估计得出的，该估计是 Katz-Tao 离散环猜想背景下的 Elekes-Rónyai 定理的类比。