UNIONS OF LINES IN

@article{Oberlin2014UNIONSOL,
  title={UNIONS OF LINES IN},
  author={Richard Oberlin},
  journal={Mathematika},
  year={2014},
  volume={62},
  pages={738-752}
}
  • R. Oberlin
  • Published 17 December 2014
  • Mathematics
  • Mathematika
We show that if a collection of lines in a vector space over a finite field has "dimension" at least 2(d-1) + beta, then its union has "dimension" at least d + beta. This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of k-planes. 
View PDF on arXiv
2 Citations
Highly Influential Citations
1
Background Citations
1
Methods Citations
1
Results Citations
1

2 Citations

Unions of lines in $\mathbb{R}^n$

We prove a conjecture of D. Oberlin on the dimension of unions of lines in R n . If d ≥ 1 is an integer, 0 ≤ β ≤ 1, and L is a set of lines in R n with Hausdorff dimension at least 2( d − 1) + β ,…

On the Sizes of Unions of Circles over Finite Fields

This paper considers unions of circles over finite fields. We generalize an approach used by Oberlin, where in place of unions of lines we consider unions of circles. First we prove that there exists…

References

SHOWING 1-10 OF 10 REFERENCES

An incidence bound for $k$-planes in $F^n$ and a planar variant of the Kakeya maximal function

We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is…

Recent work connected with the Kakeya problem

where Sn−1 is the unit sphere in R. This paper will be mainly concerned with the following issue, which is still poorly understood: what metric restrictions does the property (1) put on the set E?…

On the size of Kakeya sets in finite fields

The motivation for studying Kakeya sets over finite fields is to try to better understand the more complicated questions regarding Kakeya sets in W1. A Kakeya set K C Rn is a compact set containing a…

Restriction and Kakeya phenomena for finite fields

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this…

A sum-product estimate in finite fields, and applications

A Szemerédi-Trotter type theorem in finite fields is proved, and a new estimate for the Erdös distance problem in finite field, as well as the three-dimensional Kakeya problem in infinite fields is obtained.

The Kakeya set and maximal conjectures for algebraic varieties over finite fields

Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For…

An improved bound for Kakeya type maximal functions

Unions of hyperplanes, unions of spheres, and some related estimates

QUASIEXTREMALS FOR A RADON-LIKE TRANSFORM

Convolution with an appropriate surface measure on a paraboloid is known to define a bounded operator T from L^p(R^d) to L^q(R^d) for certain exponents p,q. By a quasiextremal for the associated…

Choosing C sufficiently large depending on the relevant implicit constants (none of which depended on C), we see that (28) contradicts the assumption (20) and so we must have

  • Choosing C sufficiently large depending on the relevant implicit constants (none of which depended on C), we see that (28) contradicts the assumption (20) and so we must have