• Corpus ID: 9784864

Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type

@article{Katz2001SomeCB,
  title={Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type},
  author={Nets Hawk Katz and Terence Tao},
  journal={arXiv: Classical Analysis and ODEs},
  year={2001}
}
  • N. Katz, T. Tao
  • Published 23 January 2001
  • Mathematics
  • arXiv: Classical Analysis and ODEs
In this paper we investigate three unsolved conjectures in geomet- ric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural δ- discretized versions of these conjectures and show that in a certain sense that these discretized versions are equivalent. 

Figures from this paper

On Falconer's Distance Set Conjecture
In this paper, using a recent parabolic restriction estimate of Tao, we obtain improved partial results in the direction of Falconer's distance set conjecture in dimensions d = 3.
On the exact Hausdorff dimension of Furstenberg-type sets
In this paper we prove some lower bounds on the Haus-dorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling
A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in $\mathbb{R}^3$
We resolve a conjecture of F¨assler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in R 3 . We do this by obtaining sharp L p bounds for
Distance Measures for Well-Distributed Sets
In this paper we investigate the Erdos/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound
On the Hausdorff dimension of pinned distance sets
  • P. Shmerkin
  • Mathematics
    Israel Journal of Mathematics
  • 2019
We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has full Hausdorff dimension for all x outside of a
A combinatorial proof of a sumset conjecture of Furstenberg
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if log r/ log s is irrational and X and Y are ×rand ×sinvariant subsets of [0, 1],
New estimates on the size of $(\alpha,2\alpha)$-Furstenberg sets
We use recent advances on the discretized sum-product problem to obtain new bounds on the Hausdorff dimension of planar (α, 2α)-Fursterberg sets. This provides a quantitative improvement to the 2α +
Slices and distances: on two problems of Furstenberg and Falconer
We survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing ×2, ×3 conjecture of H. Furstenberg in ergodic theory, and
Refined Size Estimates for Furstenberg Sets via Hausdorff Measures: A Survey of Some Recent Results
In this survey we collect and discuss some recent results on the so-called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dimH) of the sets
Some toy Furstenberg sets and projections of the four-corner Cantor set
We give lower bounds for the Hausdorff dimensions of some model Furstenberg sets. In [12] Wolff noted that the following question stems from work of Furstenberg: fix α ∈ (0, 1) and consider the class
...
...

References

SHOWING 1-10 OF 28 REFERENCES
Sums of Finite Sets
We investigate numerous cardinality questions concerning sums of finite sets. A typical problem looks like the following: if A has n elements, A + B has cn, what can we deduce about A and B? How can
Hausdorff dimension and distance sets
According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝn has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than
Extremal problems in discrete geometry
TLDR
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
On the Hausdorff dimensions of distance sets
If E is a subset of ℝn (n ≥ 1) we define the distance set of E asThe best known result on distance sets is due to Steinhaus [11], namely, that, if E ⊂ ℝn is measurable with positive n-dimensional
An improved bound on the Minkowski dimension of Besicovitch sets in R^3
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In
A bilinear approach to the restriction and Kakeya conjectures
The purpose of this paper is to investigate bilinear variants of the restriction and Kakeya conjectures, to relate them to the standard formulations of these conjectures, and to give applications of
Number Theory: New York Seminar 1991-1995
1 Sums of Four Squares.- 2 On the Number of Co-Prime-Free Sets.- 3 The Primary Role of Modular Equations.- 4 Approximation Methods in Transcendental Function Computations and Some Physical
On the Dimension of Kakeya Sets and Related Maximal Inequalities
Abstract. ((Without Abstract)).
Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets
Let μ, be a positive Radon measure with compact support in the euclidean n -space ℝ n . Introducing the Fourier transform and the averages over the spheres we can write the α-energy, 0 α n , of μ as
A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four
Abstract. ((Without abstract))
...
...