On Furstenberg’s intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

Abstract

We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$ dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg’s long-standing conjecture on the dimension of the intersections of $\times p$- and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions.

The proof of the main result is inspired by M. Hochman’s approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman’s theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

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      author = {Hochman, Michael and Shmerkin, Pablo},
      title = {Local entropy averages and projections of fractal measures},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {175},
      year = {2012},
      number = {3},
      pages = {1001--1059},
      issn = {0003-486X},
      mrclass = {28A78 (28A75 28A80)},
      mrnumber = {2912701},
      mrreviewer = {Manuel Mor\'{a}n},
      doi = {10.4007/annals.2012.175.3.1},
      url = {https://doi.org/10.4007/annals.2012.175.3.1},
      zblnumber = {1251.28008},
      }
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      pages = {521--539},
      issn = {0305-0041},
      mrclass = {11K55 (11K16 11K36)},
      mrnumber = {2838343},
      mrreviewer = {Karma Dajani},
      doi = {10.1017/S0305004111000466},
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      }
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      volume={25},
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      mrnumber = {0360498},
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      mrclass = {42B25 (28A78)},
      mrnumber = {1749437},
      mrreviewer = {Esa Järvenpää},
      doi = {10.1215/S0012-7094-00-10222-0},
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      }
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    @INCOLLECTION{PSS00,
      author = {Peres, Yuval and Schlag, Wilhelm and Solomyak, Boris},
      title = {Sixty years of {B}ernoulli convolutions},
      booktitle = {Fractal Geometry and Stochastics, {II}},
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      series = {Progr. Probab.},
      volume = {46},
      pages = {39--65},
      publisher = {Birkhäuser, Basel},
      year = {2000},
      mrclass = {42A85 (11R06 26A46 28A78 28A80)},
      mrnumber = {1785620},
      mrreviewer = {K\'{a}roly Simon},
      zblnumber = {0961.42006},
      doi = {10.1007/978-3-0348-8380-1_2},
     }
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      mrclass = {28A80 (11K60 28A78 37C45)},
      mrnumber = {2470633},
      mrreviewer = {Manuel Mor\'{a}n},
      doi = {10.1017/S0143385708000369},
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      }
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      doi = {10.4310/MRL.1996.v3.n2.a8},
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      mrreviewer = {K\'{a}roly Simon},
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    @INCOLLECTION{Shmerkin15,
      author = {Shmerkin, Pablo},
      title = {Projections of self-similar and related fractals: a survey of recent developments},
      booktitle = {Fractal Geometry and Stochastics {V}},
      series = {Progr. Probab.},
      volume = {70},
      pages = {53--74},
      publisher = {Birkhäuser/Springer, Cham},
      year = {2015},
      mrclass = {28A78 (28A80)},
      mrnumber = {3558150},
      doi = {10.1007/978-3-319-18660-3_4},
      url = {https://doi.org/10.1007/978-3-319-18660-3_4},
      zblnumber = {1338.28004},
      }
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      url = {https://doi.org/10.1017/S0305004116000335},
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      }
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    @ARTICLE{ShmerkinSolomyak16,
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      journal = {Trans. Amer. Math. Soc.},
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      fjournal = {Memoirs of the American Mathematical Society},
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      year = {2018},
      number = {1195},
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      issn = {0065-9266},
      isbn = {978-1-4704-2688-0; 978-1-4704-4264-4},
      mrclass = {60D05 (28A75 28A78 28A80 42A38 60G46 60G57)},
      mrnumber = {3756896},
      mrreviewer = {Elena Villa},
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      url = {https://doi.org/10.4007/annals.2019.189.3.2},
      }

Authors

Pablo Shmerkin

Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina