A non-linear version of Bourgain's projection theorem

被引:9
作者
Shmerkin, Pablo [1 ]
机构
[1] Univ British Columbia, Dept Math, Vancouver, BC, Canada
基金
欧洲研究理事会;
关键词
Projection theorems; distance sets; pinned distance sets; incidences; Hausdorff dimension; patterns; Lipschitz functions; HAUSDORFF DIMENSION; SETS;
D O I
10.4171/JEMS/1283
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We prove a version of Bourgain's projection theorem for parametrized families of C2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in R2 or close to 3/2 in R3, then for y E A outside of a very sparse set, the pinned distance set {lx - yl : x E A} has Hausdorff dimension at least 1/2 + c, where c is universal. Furthermore, the same holds if the distances are taken with respect to a C2 norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between 8-balls and 8-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into "Frostman pieces" that may be of independent interest.
引用
收藏
页码:4155 / 4204
页数:50
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