ARE LINES MUCH BIGGER THAN LINE SEGMENTS?

被引：5

作者：

Keleti, Tamas ^{[1
]}

机构：

[1] Eotvos Lorand Univ, Inst Math, Pazmany Peter Setany 1-C, H-1117 Budapest, Hungary

来源：

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY | 2016年 / 144卷 / 04期

关键词：

Hausdorff dimension; lines; union of line segments; Besicovitch set; Nikodym set; Kakeya Conjecture; MINKOWSKI DIMENSION; BESICOVITCH SETS; KAKEYA;

D O I：

10.1090/proc/12978

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We pose the following conjecture: (star) If A is the union of line segments in R-n, and B is the union of the corresponding full lines, then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in R-n has Hausdorff dimension at least n - 1 and (upper) Minkowski dimension n. We also prove that conjecture (star) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.

引用

页码：1535 / 1541

页数：7

共 19 条

[1]

[Anonymous], 1927, Fund. Math.

[2]

Besicovitch A., 1919, J SOC PHYS MATH, V2, P105

[3] SOME REMARKS ON KAKEYA PROBLEM [J].

DAVIES, RO .

PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY-MATHEMATICAL AND PHYSICAL SCIENCES, 1971, 69 (MAY) :417-&

[4]

Dvir Z, 2009, J AM MATH SOC, V22, P1093

[5]

Elekes M., COMMUNICATION

[6]

Falconer K. J., 1986, CAMBRIDGE TRACTS MAT, V85

[7]

Falconer Kenneth, 2014, Fractal geometry, V3rd

[8] Inhomogeneous self-similar sets and box dimensions [J].

Fraser, Jonathan M. .

STUDIA MATHEMATICA, 2012, 213 (02) :133-156

[9]

Fremlin DH., 2003, MEASURE THEORY TOPOL, V4

[10] New bounds for Kakeya problems [J].

Katz, NH ;

Tao, T .

JOURNAL D ANALYSE MATHEMATIQUE, 2002, 87 (1) :231-263

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