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Questions tagged [kakeya-sets]

Use this tag for questions relating to Kakeya and Besicovitch sets

Let $E$ be a subset of $\mathbb{R}^n$ with a unit line degment in every direction. This is a Kakeya set.

They have arbitrarily small Lebesgue measure and imply the solution to the Kakeya needle problem: What is the minimum area in the plane needed to rotate a unit line segment $360^o$?

The conjecture is trivial in one dimension, settled in two dimensions, open in dimensions $3$ and higher.

It has been connected to the areas: combinatorial geometry, additive combinatorics, multiscale analysis, heat flows, algebraic geometry, and algebraic topology.

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Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the same process but on a Besicovitch set…
Marcus Johnson
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Flat Points in Irreducible Algebraic Varieties

I am trying to understand the paper "Algebraic Methods in Discrete Analogs of the Kakeya Problem" by L. Guth and N. H. Katz. This paper contains the following lemma: Let $S$ be the set of points in $\mathbb{R}^3$ on which the polynomial $p$…
Rob F
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Variation of the Kakeya Needle Problem with positive thickness

What is the least area in the plane required to continuously rotate a needle of unit length and positive thickness $a \in ]0,1[$ around completely (i.e. $360°$)? The answer is known to be $0$ (we take the infimum) if $a=0$. An…
Alphonse
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$\{x \in \mathbb{R}^d \; : \; (x,t) \in A \; \forall t \in [0,1]\}$ is Borel whenever $A \subset \mathbb{R}^{d+1}$ is Borel.

I'm trying to show that if $E \subset \mathbb{R}^d$ is Borel, then the set $\{ (x,\omega) \in \mathbb{R}^d \times S^{d-1} \; : \; x + [0,1]\omega \subset E \}$ is Borel, where I define $x + [0,1]\omega := \{x + t \omega \; : \; t \in [0,1]\}$. My…
Paul
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If $K$ is a Kakeya set in $F_q^m$ then $K\times F_q^{n-m}$ is a Kakeya set in $F_q^n$?

Prove or disaprove the following statment: Let $q$ the power of a prime number $p$ and $n,m\in \mathbb{Z}^{+}$ such that $n
Alan Jr
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Is $K=F_q^2\setminus V(f)$ is a Kakeya set in $F_q^2$?

Statment Let $q$ the power of a prime number $p$ and $f(x,y)\in F_q[x,y]$ a non constant polynomial, then $K=F_q^2\setminus V(f)$ is a Kakeya set. My attempt. I think that it is False, for it let consider the line $L_{v,a}=\lbrace a+kv\mid k\in…
Alan Jr
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Algorithm for generating Kakeya Sets in Finite Fields

I am searching for an implementation of generating Kakeya Sets in Finite fields as defined in Dvirs proof of the finite field kakeya conjecture. I search for an implementation, which gives me precisely the points of the Kakeya set $ K \subset…
LogicTheorist
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A stronger variant of Kakeya conjecture - is it known?

The following is still unknown: Kakeya set conjecture: Define a Besicovitch set in $\mathbb{R}^n$ to be a set which contains a unit line segment in every direction. Is it true that such sets necessarily have Hausdorff dimension and Minkowski…
porton
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