Pinned distance problem, slicing measures, and local smoothing estimates
Pinned distance problem, slicing measures, and local smoothing estimates
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固定距离问题、切片测量和局部平滑估计
DOI：
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发表时间：
2017
影响因子：
1.3
通讯作者：
Bochen Liu
中科院分区：
文献类型：
--
作者：
A. Iosevich;Bochen Liu
We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$Delta^y(E) = {|x-y|:xin E},$$ we prove that for any $E, Fsubset{Bbb R}^d$, there exists a probability measure $mu_F$ on $F$ such that for $mu_F$-a.e. $yin F$,
(1) $dim_{mathcal H}(Delta^y(E))geqeta$ if $dim_{mathcal H}(E) + frac{d-1}{d+1}dim_{mathcal H}(F) > d - 1 + eta$;
(2) $Delta^y(E)$ has positive Lebesgue measure if $dim_{mathcal H}(E)+frac{d-1}{d+1}dim_{mathcal H}(F) > d$;
(3) $Delta^y(E)$ has non-empty interior if $dim_{mathcal H}(E)+frac{d-1}{d+1}dim_{mathcal H}(F) > d+1$.
We also show that in the case when $dim_{mathcal H}(E)+frac{d-1}{d+1}dim_{mathcal H}(F)>d$, for $mu_F$-a.e. $yin F$, $$ left{tin{Bbb R} : dim_{mathcal H}({xin E:|x-y|=t}) geq dim_{mathcal H}(E)+frac{d+1}{d-1}dim_{mathcal H}(F)-d
ight} $$ has positive Lebesgue measure. This describes dimensions of slicing subsets of $E$, sliced by spheres centered at $y$.
In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.
We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$Delta^y(E) = {|x-y|:xin E},$$ we prove that for any $E, Fsubset{Bbb R}^d$, there exists a probability measure $mu_F$ on $F$ such that for $mu_F$-a.e. $yin F$,
(1) $dim_{mathcal H}(Delta^y(E))geqeta$ if $dim_{mathcal H}(E) + frac{d-1}{d+1}dim_{mathcal H}(F) > d - 1 + eta$;
(2) $Delta^y(E)$ has positive Lebesgue measure if $dim_{mathcal H}(E)+frac{d-1}{d+1}dim_{mathcal H}(F) > d$;
(3) $Delta^y(E)$ has non-empty interior if $dim_{mathcal H}(E)+frac{d-1}{d+1}dim_{mathcal H}(F) > d+1$.
We also show that in the case when $dim_{mathcal H}(E)+frac{d-1}{d+1}dim_{mathcal H}(F)>d$, for $mu_F$-a.e. $yin F$, $$ left{tin{Bbb R} : dim_{mathcal H}({xin E:|x-y|=t}) geq dim_{mathcal H}(E)+frac{d+1}{d-1}dim_{mathcal H}(F)-d
ight} $$ has positive Lebesgue measure. This describes dimensions of slicing subsets of $E$, sliced by spheres centered at $y$.
In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.