Hausdorff dimension and projections related to intersections

• Autores: Pertti Mattila
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 1, 2022, págs. 305-323
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • For Sg(x, y) = x − g(y), x, y ∈ Rn, g ∈ O(n), we investigate the Lebesguemeasure and Hausdorff dimension of Sg(A) given the dimension of A, both for generalBorel subsets of R2n and for product sets.

• Referencias bibliográficas
  • A. Barron, M. B. Erdogan, and T. L. J. Harris, Fourier decay of fractal measures on hyperboloids, Trans. Amer. Math. Soc. 374(2) (2021), 1041–1075....
  • C.-H. Cho, S. Ham, and S. Lee, Fractal Strichartz estimate for the wave equation, Nonlinear Anal. 150 (2017), 61–75. DOI: 10.1016/j.na.2016.11.006.
  • X. Du, Upper bounds for Fourier decay rates of fractal measures, J. Lond. Math. Soc. (2) 102(3) (2020), 1318–1336. DOI: 10.1112/jlms.12364.
  • X. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, and R. Zhang, Weighted restriction estimates and application to Falconer distance set problem,...
  • X. Du, A. Iosevich, Y. Ou, H. Wang, and R. Zhang, An improved result for Falconer’s distance set problem in even dimensions, Math. Ann. 380(3–4)...
  • X. Du and R. Zhang, Sharp L2 estimates of the Schr¨odinger maximal function in higher dimensions, Ann. of Math. (2) 189(3) (2019), 837–861....
  • M. B. Erdogan, A note on the Fourier transform of fractal measures, Math. Res. Lett. 11(2–3) (2004), 299–313. DOI: 10.4310/MRL.2004.v11.n3.a3.
  • S. Eswarathasan, A. Iosevich, and K. Taylor, Intersections of sets and Fourier analysis, J. Anal. Math. 128 (2016), 159–178. DOI: 10.1007/s11854-016...
  • K. J. Falconer, Hausdorff dimension and the exceptional set of projections, Mathematika 29(1) (1982), 109–115. DOI: 10.1112/S0025579300012201.
  • K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32(2) (1985), 206–212 (1986). DOI: 10.1112/S0025579300010998. [11]...
  • L. Guth, A. Iosevich, Y. Ou, and H. Wang, On Falconer’s distance set problem in the plane, Invent. Math. 219(3) (2020), 779–830. DOI: 10.1007/...
  • T. L. J. Harris, Improved decay of conical averages of the Fourier transform, Proc. Amer. Math. Soc. 147(11) (2019), 4781–4796. DOI: 10.1090/proc/14747.
  • T. L. J. Harris, Restricted families of projections onto planes: the general case of nonvanishing geodesic curvature, Preprint (2021). arXiv:2107.14701.
  • A. Iosevich and B. Liu, Falconer distance problem, additive energy and Cartesian products, Ann. Acad. Sci. Fenn. Math. 41(2) (2016), 579–585....
  • E. Järvenpää, M. Järvenpää, and T. Keleti ¨ , Hausdorff dimension and nondegenerate families of projections, J. Geom. Anal. 24(4) (2014),...
  • E. Järvenpää, M. Järvenpää, F. Ledrappier, and M. Leikas, One-dimensional families of projections, Nonlinearity 21(3) (2008), 453–463. DOI:...
  • M. Järvenpää, On the upper Minkowski dimension, the packing dimension, and orthogonal projections, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes...
  • M. Järvenpää, Concerning the packing dimension of intersection measures, Math. Proc. Cambridge Philos. Soc. 121(2) (1997), 287–296. DOI: 10.1017/...
  • M. Järvenpää, Packing dimension, intersection measures, and isometries, Math. Proc. Cambridge Philos. Soc. 122(3) (1997), 483–490. DOI: 10.1017/...
  • A. Käenmäki, T. Orponen, and L. Venieri, A Marstrand-type restricted projection theorem in R3, Preprint (2017). arXiv:1708.04859.
  • J.-P. Kahane, Sur la dimension des intersections, in: “Aspects of Mathematics and its Applications”, North-Holland Math. Library 34, North-Holland,...
  • R. Kaufman, On Hausdorff dimension of projections, Mathematika 15(2) (1968), 153–155. DOI: 10.1112/S0025579300002503.
  • R. Kaufman and P. Mattila, Hausdorff dimension and exceptional sets of linear transformations, Ann. Acad. Sci. Fenn. Ser. A I Math. 1(2) (1975),...
  • J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4(1) (1954),...
  • P. Mattila, Hausdorff dimension and capacities of intersections of sets in nspace, Acta Math. 152(1–2) (1984), 77–105. DOI: 10.1007/BF02392192.
  • P. Mattila, On the Hausdorff dimension and capacities of intersections, Mathematika 32(2) (1985), 213–217 (1986). DOI: 10.1112/S0025579300011001.
  • P. Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika...
  • P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces”, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44,...
  • P. Mattila, “Fourier Analysis and Hausdorff Dimension”, Cambridge Studies in Advanced Mathematics 150, Cambridge University Press, Cambridge,...
  • P. Mattila, Exceptional set estimates for the Hausdorff dimension of intersections, Ann. Acad. Sci. Fenn. Math. 42(2) (2017), 611–620. DOI:...
  • P. Mattila, Hausdorff Dimension, Projections, Intersections, and Besicovitch Sets, in: “New Trends in Applied Harmonic Analysis”, Volume 2,...
  • P. Mattila, Hausdorff dimension of intersections with planes and general sets, Preprint (2020). arXiv:2005.11790.
  • D. M. Oberlin, Exceptional sets of projections, unions of k-planes and associated transforms, Israel J. Math. 202(1) (2014), 331–342. DOI:...
  • D. Oberlin and R. Oberlin, Application of a Fourier restriction theorem to certain families of projections in R3, J. Geom. Anal. 25(3) (2015),...
  • T. Orponen, Hausdorff dimension estimates for restricted families of projections in R3, Adv. Math. 275 (2015), 147–183. DOI: 10.1016/j.aim.2015.02.011.
  • T. Orponen and L. Venieri, Improved bounds for restricted families of projections to planes in R3, Int. Math. Res. Not. IMRN 2020(19) (2020),...
  • T. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 1999(10) (1999), 547–567. DOI: 10.1155/S1073792899...