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

@inproceedings{Shmerkin2021OnTD, title={On the distance sets spanned by sets of dimension \$d/2\$ in \$\mathbb\{R\}^d\$}, author={Pablo Shmerkin and Hong Wang}, year={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 dimensions d = 2 or 3, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension d/2; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension 1 has Hausdorff dimension at least ( √ 5 − 1)/2 ≈ 0.618. In higher…

## 4 Citations

### Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem

- Mathematics
- 2022

A BSTRACT . We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let X, Y Ă R 2 be non-empty Borel sets. If X is not contained on any line,…

### On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

- Mathematics
- 2021

Let 0 ď s ď 1 and 0 ď t ď 2. An ps, tq-Furstenberg set is a setK Ă R with the following property: there exists a line set L of Hausdorff dimension dimH L ě t such that dimHpK X `q ě s for all ` P L.…

### On exceptional sets of radial projections

- Mathematics
- 2022

A BSTRACT . We prove two new exceptional set estimates for radial projections in the plane. If K Ă R 2 is a Borel set with dim H K ą 1 , then dim H t x P R 2 z K : dim H π x p K q ď σ u ď max t 1 ` σ…

### Discretized sum-product type problems: Energy variants and Applications

- Mathematics, Computer Science
- 2022

This paper establishes non-trivial estimates for the additive discretized energy of X c ∈ C, and proves new explicit upper bounds on the quantity dim H, which leads to considerably shorter proofs over the previous works due to Bourgain and Orponen.

## References

SHOWING 1-10 OF 40 REFERENCES

### Improved bounds for the dimensions of planar distance sets

- MathematicsJournal of Fractal Geometry
- 2020

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…

### 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…

### Distance sets, orthogonal projections and passing to weak tangents

- MathematicsIsrael Journal of Mathematics
- 2018

We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve…

### On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

- Mathematics
- 2021

Let 0 ď s ď 1 and 0 ď t ď 2. An ps, tq-Furstenberg set is a setK Ă R with the following property: there exists a line set L of Hausdorff dimension dimH L ě t such that dimHpK X `q ě s for all ` P L.…

### 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 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…

### A nonlinear projection theorem for Assouad dimension and applications

- MathematicsJournal of the London Mathematical Society
- 2022

We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum‐product phenomena. In the setting…

### On restricted families of projections in ℝ3

- Mathematics
- 2014

We study projections onto non‐degenerate one‐dimensional families of lines and planes in R3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension…

### 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…