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Distance sets, orthogonal projections and passing to weak tangents

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Abstract

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 the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.

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References

  1. P. Assouad. Espaces métriques, plongements, facteurs, Thèse de doctorat d’État, Publications Mathématiques d’Orsay, 223–7769, Université Paris XI, Orsay, 1977.

    MATH  Google Scholar 

  2. C. J. Bishop and Y. Peres. Fractals in Probability and Analysis, Cambridge Studies in Advanced Mathematics, Vol. 162, Cambridge University Press, Cambridge, 2017.

  3. M. G. Bouligand. Ensembles impropres et nombre dimensionnel, Bulletin des Sciences Mathématiques 52 (1928), 320–344, 361–376.

    MATH  Google Scholar 

  4. J. Bourgain. The discretized sum-product and projection theorems, Journal d’Analyse Mathématique 112 (2010), 193–236.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. O. Davies, J. M. Marstrand and S. J. Taylor. On the intersections of transforms of linear sets, Colloquium Mathematicum 7 (1959/1960), 237–243.

    Article  MathSciNet  Google Scholar 

  6. K. J. Falconer. On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), 206–212.

    Article  MathSciNet  MATH  Google Scholar 

  7. K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 2014.

    Google Scholar 

  8. K. J. Falconer, J. M. Fraser and X. Jin. Sixty years of fractal projections, in Fractal Geometry and Stochastics V, Progress in Probability, Vol. 70, Birkhäuser/Springer, Cham, 2015, pp. 3–25.

    MathSciNet  MATH  Google Scholar 

  9. K. Fässler and T. Orponen. On restricted families of projections in R3, Proceedings of the London Mathematical Society 109 (2014), 353–381.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Ferguson, J. M. Fraser and T. Sahlsten. Scaling scenery of (×m,×n) invariant measures, Advances in Mathematics 268 (2015), 564–602.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. M. Fraser. Assouad type dimensions and homogeneity of fractals, Transactions of the American Mathematical Society 366 (2014), 6687–6733.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. M. Fraser and T. Orponen. The Assouad dimensions of projections of planar sets, Proceedings of the London Mathematical Society 114 (2017), 374–398.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. M. Fraser and H. Yu. Arithmetic patches, weak tangents, and dimension, Bulletin of the London Mathematical Society 50 (2018), 85–95.

    Article  Google Scholar 

  14. H. Furstenberg. Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis. A Symposium in Honor of Salomon Bochner, Princeton Mathematical Series, Vol. 31, Princeton University Press, Princeton, NJ, 1970, pp. 41–59.

    Google Scholar 

  15. H. Furstenberg. Ergodic fractal measures and dimension conservation, Ergodic Theory and Dynamical Systems 28 (2008), 405–422.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Hochman. Dynamics on fractals and fractal distributions, preprint, (2010), available at https://doi.org/arxiv.org/abs/1008.3731.

    Google Scholar 

  17. M. Hochman and P. Shmerkin. Local entropy averages and projections of fractal measures, Annals of Mathematics 175 (2012), 1001–1059.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Käenmäki, T. Ojala and E. Rossi. Rigidity of quasisymmetric mappings on selfaffine carpets, Internationa Mathematics Research Notices, https://doi.org/10.1093/imrn/rnw336.

  19. A. Käenmäki and E. Rossi, Weak separation condition, Assouad dimension, and Furstenberg homogeneity, Annales Academiæ Scientiarum Fennicæ. Mathematica 41 (2016), 465–490.

    MATH  Google Scholar 

  20. T. Keleti, D. T. Nagy and P. Shmerkin, Squares and their centers, Journal d’Analyse Mathématique 134 (2018), 643–669.

    Article  MathSciNet  Google Scholar 

  21. E. Le Donne and T. Rajala, Assouad dimension, Nagata dimension, and uniformly close metric tangents, Indiana University Mathematics Journal 64 (2015), 21–54.

    Article  MATH  Google Scholar 

  22. J. Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, Journal of the Korean Mathematical Society 35 (1998), 23–76.

    MathSciNet  MATH  Google Scholar 

  23. J. M. Mackay and J. T. Tyson, Conformal Dimension. Theory and Application, University Lecture Series, Vol. 54, American Mathematical Society, Providence, RI, 2010.

    Google Scholar 

  24. J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proceedings of the London Mathematical Society 4 (1954), 257–302.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Annales Academiæ Scientiarum Fennicæ. Series A I. Mathematica 1 (1975), 227–244.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  27. P. Mattila, Recent progress on dimensions of projections, in Geometry and Analysis of Fractals, Springer Proceedings in Mathematics & Statistics, Vol. 88, Springer-Verlag, Berlin–Heidelberg, 2014, pp. 283–301.

    Google Scholar 

  28. P. Mattila, Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, Vol. 150, Cambridge University Press, Cambridge, 2015.

    Google Scholar 

  29. T. Orponen, On the distance sets of self-similar sets, Nonlinearity 25 (2012), 1919–1929.

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Orponen, On the packing dimension and category of exceptional sets of orthogonal projections, Annali di Matematica Pura ed Applicata 194 (2015), 843–880.

    Article  MathSciNet  MATH  Google Scholar 

  31. T. Orponen, On the distance sets of Ahlfors–David regular sets, Advances in Mathematics 307 (2017), 1029–1045.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Orponen, An improved bound on the packing dimension of Furstenberg sets in the plane, Journal of the European Mathematical Society, to appear.

  33. J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, Vol. 186, Cambridge University Press, Cambridge, 2011.

    MATH  Google Scholar 

  34. P. Shmerkin, On distance sets, box-counting and Ahlfors-regular sets, Discrete Analysis 9 (2017).

    Google Scholar 

  35. P. Shmerkin, On the Hausdorff dimension of pinned distance sets, preprint, (2017), available at https://doi.org/arxiv.org/abs/1706.00131.

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of St Andrews, St Andrews, KY16 9SS, Scotland

    Jonathan M. Fraser

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  1. Jonathan M. Fraser
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Correspondence to Jonathan M. Fraser.

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Fraser, J.M. Distance sets, orthogonal projections and passing to weak tangents. Isr. J. Math. 226, 851–875 (2018). https://doi.org/10.1007/s11856-018-1715-z

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  • DOI: https://doi.org/10.1007/s11856-018-1715-z

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