# On the Hausdorff dimension of pinned distance sets

@article{Shmerkin2017OnTH, title={On the Hausdorff dimension of pinned distance sets}, author={Pablo Shmerkin}, journal={Israel Journal of Mathematics}, year={2017}, volume={230}, pages={949-972} }

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 set of Hausdorff dimension 1 (in particular, for many x ∈ A). This verifies a strong variant of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint s = 1.

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## References

SHOWING 1-10 OF 24 REFERENCES

### 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…

### Sums of Cantor sets yielding an interval

- MathematicsJournal of the Australian Mathematical Society
- 2002

Abstract In this paper we prove that if a Cantor set has ratios of dissection bounded away from zero, then there is a natural number N, such that its N-fold sum is an interval. Moreover, for each…

### Local entropy averages and projections of fractal measures

- Mathematics
- 2012

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure…

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

- Mathematics
- 2001

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…

### Hausdorff dimension, intersections of projections and exceptional plane sections

- Mathematics
- 2016

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion…

### Resonance between Cantor sets

- MathematicsErgodic Theory and Dynamical Systems
- 2009

Abstract Let Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two…

### SCALING SCENERY OF ( × m , × n ) INVARIANT MEASURES

- Mathematics
- 2018

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x, y) 7→ (mx mod 1, ny mod 1) that are Bernoulli measures for the natural Markov…

### On distance sets, box-counting and Ahlfors-regular sets

- Mathematics
- 2016

We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent $s>1$. As a corollary, we improve upon a recent result of Orponen,…

### Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions

- Mathematics
- 2000

Erdős (1939, 1940) studied the distribution νλ of the random series P∞ 0 ±λn, and showed that νλ is singular for infinitely many λ ∈ (1/2, 1), and absolutely continuous for a.e. λ in a small interval…