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 the distance set problem in geometric measure theory introduced by K. Falconer. We discuss some of the ideas behind our solution of Furstenberg’s slicing conjecture, and recent progress in Falconer’s problem. While these two problems are on the surface rather different, we emphasize some common themes in our approach: analyzing fractals through a combinatorial description in terms of “branching numbers”, and viewing the problems through a “multiscale projection” lens.

[1]  H. Furstenberg.  Intersections of Cantor Sets and Transversality of Semigroups , 2015 .

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

[4]  J. Zahl,et al.  On the Discretized Sum-Product Problem , 2018, International Mathematics Research Notices.

[5]  Misha Rudnev,et al.  An update on the sum-product problem , 2021, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[7]  Joshua Zahl,et al.  An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³ , 2017, Journal of the American Mathematical Society.

[8]  Y. Peres,et al.  Convolutions of cantor measures without resonance , 2009, 0905.3850.

[9]  Tuomas Orponen,et al.  On arithmetic sums of Ahlfors-regular sets , 2021, Geometric and Functional Analysis.

[10]  Jean Bourgain,et al.  Hausdorff dimension and distance sets , 1994 .

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

[12]  T. Wolff.  Decay of circular means of Fourier transforms of measures , 1999 .

[13]  Jean Bourgain,et al.  On the spectral gap for finitely-generated subgroups of SU(2) , 2007 .

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

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

[16]  M. Hochman.  DIMENSION THEORY OF SELF-SIMILAR SETS AND MEASURES , 2019, Proceedings of the International Congress of Mathematicians (ICM 2018).

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

[18]  Y. Peres,et al.  Resonance between Cantor sets , 2007, Ergodic Theory and Dynamical Systems.

[19]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[20]  Tuomas Orponen.  On the Assouad dimension of projections , 2019, Proceedings of the London Mathematical Society.

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

[22]  B. Adamczewski,et al.  Mahler's method in several variables and finite automata , 2020, 2012.08283.

[23]  P. Mattila.  Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets , 1987 .

[24]  Harry Furstenberg,et al.  Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation , 1967, Mathematical systems theory.

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

[26]  Michael Hochman,et al.  On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$ , 2012, 1503.09043.

[27]  A. Algom.  Slicing theorems and rigidity phenomena for self‐affine carpets , 2018, Proceedings of the London Mathematical Society.

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

[29]  Larry Guth,et al.  Weighted restriction estimates and application to Falconer distance set problem , 2018, American Journal of Mathematics.

[30]  Joshua Zahl,et al.  On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-R\'onyai theorem , 2021 .

[31]  H. Furstenberg.  Ergodic fractal measures and dimension conservation , 2008, Ergodic Theory and Dynamical Systems.

[32]  Digit expansions of numbers in different bases , 2019, 1905.00832.

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

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

[35]  On the dimension of Bernoulli convolutions for all transcendental parameters , 2018, Annals of Mathematics.

[36]  Tim Austin.  A new dynamical proof of the Shmerkin–Wu theorem , 2020, Journal of Modern Dynamics.

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

[38]  Additive and geometric transversality of fractal sets in the integers , 2020, 2007.05480.

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

[40]  P. Shmerkin,et al.  On measures that improve $L^q$ dimension under convolution , 2018, Revista Matemática Iberoamericana.

[41]  Gerald A. Edgar,et al.  Borel subrings of the reals , 2002 .

[42]  Meng Wu.  A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets , 2016, Annals of Mathematics.

[43]  P. Shmerkin,et al.  On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions , 2016, Annals of Mathematics.

[44]  J. Bourgain,et al.  Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus , 2011 .

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

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

[47]  M. Burak Erdo˜gan.  A Bilinear Fourier Extension Theorem and Applications to the Distance Set Problem , 2005 .

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

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

[50]  Amir Algom,et al.  Improved versions of some Furstenberg type slicing Theorems for self-affine carpets , 2021, 2107.02068.

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

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

[53]  Ka-Sing Lau,et al.  Multifractal Measures and a Weak Separation Condition , 1999 .