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

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 dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension d/2. These results rely on new estimates for the dimensions of radial projections that may have independent interest.

[1]  M. Hochman,et al.  Local entropy averages and projections of fractal measures , 2009, 0910.1956.

[2]  Pablo Shmerkin.  Improved bounds for the dimensions of planar distance sets , 2018, Journal of Fractal Geometry.

[3]  K. Falconer.  On the Hausdorff dimensions of distance sets , 1985 .

[4]  Weikun He.  Orthogonal projections of discretized sets , 2017, Journal of Fractal Geometry.

[5]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[6]  Distance sets, orthogonal projections and passing to weak tangents , 2017, Israel Journal of Mathematics.

[7]  L. Guth,et al.  On the Erdős distinct distances problem in the plane , 2015 .

[8]  K. Falconer,et al.  Packing dimensions of sections of sets , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  M. Mirzakhani,et al.  Introduction to Ergodic theory , 2010 .

[10]  J. Fraser.  Assouad Dimension and Fractal Geometry , 2020, 2005.03763.

[11]  L. Guth,et al.  On Falconer’s distance set problem in the plane , 2018, Inventiones mathematicae.

[12]  Y. Peres,et al.  Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions , 2000 .

[13]  Bochen Liu.  Hausdorff dimension of pinned distance sets and the 𝐿²-method , 2018, Proceedings of the American Mathematical Society.

[14]  Shashi M. Srivastava,et al.  A Course on Borel Sets , 1998, Graduate texts in mathematics.

[15]  R. Kaufman.  On Hausdorff dimension of projections , 1968 .

[16]  Terence Tao,et al.  Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type , 2001 .

[18]  Ruixiang Zhang,et al.  Sharp L2 estimates of the Schrödinger maximal function in higher dimensions , 2018, Annals of Mathematics.

[19]  J. Bourgain.  On the Erdős-Volkmann and Katz-Tao ring conjectures , 2003 .

[20]  Tuomas Orponen.  On the distance sets of Ahlfors–David regular sets , 2015, 1509.06675.

[21]  Tuomas Orponen,et al.  On restricted families of projections in ℝ3 , 2013, 1302.6550.

[22]  P. Shmerkin.  On the Hausdorff dimension of pinned distance sets , 2017, Israel Journal of Mathematics.

[23]  A nonlinear projection theorem for Assouad dimension and applications , 2020, 2004.12001.

[24]  Noam Solomon,et al.  Incidence Estimates for Well Spaced Tubes , 2019, Geometric and Functional Analysis.

[25]  Tuomas Orponen.  On the dimension and smoothness of radial projections , 2017, Analysis & PDE.

[26]  P. Shmerkin,et al.  A non-linear version of Bourgain’s projection theorem , 2020, Journal of the European Mathematical Society.

[27]  P. Mattila.  Fourier Analysis and Hausdorff Dimension , 2015 .

[28]  Pablo Shmerkin,et al.  On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane , 2021, 2106.03338.

[29]  J. Bourgain.  The discretized sum-product and projection theorems , 2010 .

[30]  Bochen Liu.  An L2-identity and pinned distance problem , 2018, Geometric and Functional Analysis.

[31]  P. Shmerkin,et al.  New Bounds on the Dimensions of Planar Distance Sets , 2018, Geometric and Functional Analysis.