# Improved bounds for the dimensions of planar distance sets

@article{Shmerkin2018ImprovedBF, title={Improved bounds for the dimensions of planar distance sets}, author={Pablo Shmerkin}, journal={Journal of Fractal Geometry}, year={2018} }

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if $A$ has Hausdorff dimension $>1$, then the set of distances spanned by points of $A$ has Hausdorff dimension at least $40/57 > 0.7$ and there are many $y\in A$ such that the pinned distance set $\{ |x-y|:x\in A\}$ has Hausdorff…

## 15 Citations

### On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$

- Mathematics
- 2021

We establish the dimension version of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In…

### A non-linear version of Bourgain’s projection theorem

- MathematicsJournal of the European Mathematical Society
- 2022

We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a…

### Finite point configurations and the regular value theorem in a fractal setting

- MathematicsIndiana University Mathematics Journal
- 2022

In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue…

### Hausdorff dimension of pinned distance sets and the $L^2$-method

- Mathematics
- 2018

. We prove that for any compact set E ⊂ R 2 , dim H ( E ) > 1, there exists x ∈ E such that the Hausdorﬀ dimension of the pinned distance set ∆ x ( E ) = {| x − y | : y ∈ E } is no less than min…

### Structural theorems on the distance sets over finite fields

- Mathematics
- 2021

Let F q be a ﬁnite ﬁeld of order q . Iosevich and Rudnev (2007) proved that for any set A ⊂ F dq , if | A | ≫ q d +12 , then the distance set ∆( A ) contains a positive proportion of all distances.…

### Pinned Distance Sets Using Effective Dimension

- MathematicsArXiv
- 2022

In this paper, we use algorithmic tools, eﬀective dimension and Kolmogorov complexity, to study the fractal dimension of distance sets. We show that, for any analytic set E ⊆ R 2 of Hausdorﬀ…

### On the Mattila–Sjölin distance theorem for product sets

- MathematicsMathematika
- 2022

Let A be a compact set in R$\mathbb {R}$ , and E=Ad⊂Rd$E=A^d\subset \mathbb {R}^d$ . We know from the Mattila–Sjölin's theorem if dimH(A)>d+12d$\dim _H(A)>\frac{d+1}{2d}$ , then the distance set…

### Low-Dimensional Pinned Distance Sets Via Spherical Averages

- Mathematics, Computer ScienceThe Journal of Geometric Analysis
- 2021

An inequality is derived for the average energies of pinned distance measures that gives an analogue of Liu’s theorem for pinned distance sets of dimension smaller than 1.

### Slices and distances: on two problems of Furstenberg and Falconer

- Mathematics
- 2021

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…

### On the multiparameter Falconer distance problem

- Mathematics
- 2021

We study an extension of the Falconer distance problem in the multiparameter setting. Given l ě 1 and R “ R1 ˆ ̈ ̈ ̈ ˆ Rl , di ě 2. For any compact set E Ă R with Hausdorff dimension larger than d…

## References

SHOWING 1-10 OF 12 REFERENCES

### On the dimension and smoothness of radial projections

- MathematicsAnalysis & PDE
- 2019

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces.
To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$…

### On the Hausdorff dimension of pinned distance sets

- MathematicsIsrael 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…

### Pinned distance problem, slicing measures, and local smoothing estimates

- MathematicsTransactions of the American Mathematical Society
- 2018

We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\Delta^y(E) = \{|x-y|:x\in E\},$$ we prove that for any…

### Hausdorff dimension of pinned distance sets and the $L^2$-method

- Mathematics
- 2018

. We prove that for any compact set E ⊂ R 2 , dim H ( E ) > 1, there exists x ∈ E such that the Hausdorﬀ dimension of the pinned distance set ∆ x ( E ) = {| x − y | : y ∈ E } is no less than min…

### Sharp L2 estimates of the Schrödinger maximal function in higher dimensions

- MathematicsAnnals of Mathematics
- 2019

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by…

### On Falconer’s distance set problem in the plane

- MathematicsInventiones mathematicae
- 2019

If $$E \subset \mathbb {R}^2$$ E ⊂ R 2 is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point $$x \in E$$ x ∈ E so that the set of distances $$\{ |x-y| \}_{y \in…

### On the visibility of planar sets

- Mathematics
- 2017

Assume that $E,K \subset \mathbb{R}^{2}$ are Borel sets with $\dim_{\mathrm{H}} K > 0$. Is a positive dimensional part of $K$ visible from some point in $E$? Not necessarily, since $E$ can be…

### On the Hausdorff dimensions of distance sets

- Mathematics
- 1985

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…

### New Bounds on the Dimensions of Planar Distance Sets

- Mathematics, Computer ScienceGeometric and Functional Analysis
- 2019

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath}…

### Weighted restriction estimates and application to Falconer distance set problem

- MathematicsAmerican Journal of Mathematics
- 2021

abstract:We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the…