scholarly journals A Generalization of the Convex Kakeya Problem

2012 ◽  
pp. 1-12
Author(s):  
Hee-Kap Ahn ◽  
Sang Won Bae ◽  
Otfried Cheong ◽  
Joachim Gudmundsson ◽  
Takeshi Tokuyama ◽  
...  
Keyword(s):  
Kakeya Problem

Das endliche Kakeya-Problem

2018 ◽  
pp. 275-280
Author(s):  
Martin Aigner ◽  
Günter M. Ziegler
Keyword(s):  
Kakeya Problem

scholarly journals Algebraic methods in discrete analogs of the Kakeya problem

2010 ◽  
Vol 225 (5) ◽  
pp. 2828-2839 ◽  
Author(s):  
Larry Guth ◽  
Nets Hawk Katz
Keyword(s):  
Algebraic Methods ◽  
Kakeya Problem ◽  
Discrete Analogs

A Remark on the Kakeya Problem

1963 ◽  
Vol 70 (7) ◽  
pp. 706
Author(s):  
A. A. Blank
Keyword(s):  
Kakeya Problem

Some remarks on the Kakeya problem

1971 ◽  
Vol 69 (3) ◽  
pp. 417-421 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Unit Circle ◽  
Small Area ◽  
Original Position ◽  
Connected Subset ◽  
Kakeya Problem ◽  
Straight Line ◽  
Small Measure ◽  
Remarkable Result ◽  
Unit Segment ◽  
Simply Connected

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.

The inverse Kakeya problem

2021 ◽  
Author(s):  
Sergio Cabello ◽  
Otfried Cheong ◽  
Michael Gene Dobbins
Keyword(s):  
Kakeya Problem

scholarly journals Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions

2017 ◽  
Vol 2019 (14) ◽  
pp. 4419-4430 ◽  
Author(s):  
Jonathan M Fraser ◽  
Kota Saito ◽  
Han Yu
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
The Other ◽  
Kakeya Problem ◽  
Higher Dimensional ◽  
Finite Arithmetic

AbstractWe provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions (APs). More precisely, we say $F$ uniformly avoids APs of length $k \geq 3$ if there is an $\epsilon>0$ such that one cannot find an AP of length $k$ and gap length $\Delta>0$ inside the $\epsilon \Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a “reverse Kakeya problem:” we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates APs in every direction.

scholarly journals On the finite field Kakeya problem in two dimensions

2007 ◽  
Vol 124 (1) ◽  
pp. 248-257 ◽  
Author(s):  
X.W.C. Faber
Keyword(s):  
Finite Field ◽  
Two Dimensions ◽  
Kakeya Problem

scholarly journals A Generalization of the Convex Kakeya Problem

Algorithmica ◽  
2013 ◽  
Vol 70 (2) ◽  
pp. 152-170 ◽  
Author(s):  
Hee-Kap Ahn ◽  
Sang Won Bae ◽  
Otfried Cheong ◽  
Joachim Gudmundsson ◽  
Takeshi Tokuyama ◽  
...  
Keyword(s):  
Kakeya Problem
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