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On arithmetic sums of fractal sets in \(\mathbb{R}^d\). (English) Zbl 1475.28002

Let \(E\subset\mathbb{R}^d\) be a nonempty compact set and denote by \(\oplus_n E:= \{x_1 + \cdots x_n: x_i\in E, i\in\{1, \ldots,n \}\}\) its \(n\)-fold aritmetic sum. \(E\) is called arithmetically thick if there exists an \(n\in\mathbb{N}\) such that the \(n\)-fold aritmetic sum of \(E\) has nonempty interior. A compact set \(E\subset\mathbb{R}^d\) is called uniformly non-flat if there exists an \(\epsilon_0>0\) such that for any \(x\in E\) and \(0 < r \leq \mathrm{diam }E\), the intersection \(E\cap B(x,r)\) never stays \(\epsilon_0 r\)- close to a hyperplane in \(\mathbb{R}^d\). The authors prove that \(E\) is arithmetically thick if \(E\) is uniformally non-flat. In addition, arithmetic thickness is shown for classes of fractal sets in \(\mathbb{R}^d\) with \(d\geq 2\) such as self-similar and self-conformal sets, and self-affine sets in \(\mathbb{R}^2\) that do not lie in a hyperplane.

MSC:

28A75 Length, area, volume, other geometric measure theory
28A80 Fractals
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[1] S.Astels, ‘Cantor sets and numbers with restricted partial quotients’, Trans. Amer. Math. Soc.352 (2000) 133-170. · Zbl 0967.11026
[2] T.Banakh, E.Jabłońska and W.Jabłoński, ‘The continuity of additive and convex functions which are upper bounded on non‐flat continua in \(\mathbb{R}^n\)’, Topol. Methods Nonlinear Anal.54 (2019) 247-256. · Zbl 1428.26022
[3] C. J.Bishop and P. W.Jones, ‘Wiggly sets and limit sets’, Ark. Mat.35 (1997) 201-224. · Zbl 0939.30031
[4] R.Broderick, L.Fishman, D.Kleinbock, A.Reich and B.Weiss, ‘The set of badly approximable vectors is strongly \(C^1\) incompressible’, Math. Proc. Cambridge Philos. Soc.153 (2012) 319-339. · Zbl 1316.11064
[5] C. A.Cabrelli, K. E.Hare and U. M.Molter, ‘Sums of Cantor sets’, Ergodic Theory Dynam. Systems17 (1997) 1299-1313. · Zbl 0891.28001
[6] C. A.Cabrelli, K. E.Hare and U. M.Molter, ‘Sums of Cantor sets yielding an interval’, J. Aust. Math. Soc.73 (2002) 405-418. · Zbl 1019.28002
[7] B. J.Conway, Functions of one complex variable II, Graduate Texts in Mathematics 159 (Springer, New York, NY, 1995). · Zbl 0887.30003
[8] G.David, ‘Hausdorff dimension of uniformly non flat sets with topology’, Publ. Mat.48 (2004) 187-225. · Zbl 1065.49028
[9] G.David and S.Semmes, Fractured fractals and broken dreams: self‐similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications 7 (Oxford University Press, New York, NY, 1997). · Zbl 0887.54001
[10] K. J.Falconer, Fractal geometry: mathematical foundations and applications (Wiley, Hoboken, NJ, 2003). · Zbl 1060.28005
[11] J. M.Fraser, D. C.Howroyd and H.Yu, ‘Dimension growth for iterated sumsets’, Math. Z.293 (2019) 1015-1042. · Zbl 1429.28013
[12] H.Furstenberg, ‘Ergodic fractal measures and dimension conservation’, Ergodic Theory Dynam. Systems28 (2008) 405-422. · Zbl 1154.37322
[13] M. J.Hall, ‘On the sum and product of continued fractions’, Ann. of Math. (2)48 (1947) 966-993. · Zbl 0030.02201
[14] J. E.Hutchinson, ‘Fractals and self‐similarity’, Indiana Univ. Math. J.30 (1981) 713-747. · Zbl 0598.28011
[15] P. W.Jones, ‘Rectifiable sets and the traveling salesman problem’, Invent. Math.102 (1990) 1-15. · Zbl 0731.30018
[16] A.Käenmäki, ‘On the geometric structure of the limit set of conformal iterated function systems’, Publ. Mat.47 (2003) 133-141. · Zbl 1036.28007
[17] H.Kestelman, ‘On the functional equation \(f ( x + y ) = f ( x ) + f ( y )\)’, Fund. Math.34 (1947) 144-147. · Zbl 0032.02804
[18] B.Lemmens and R.Nussbaum, Nonlinear Perron‐Frobenius theory, Cambridge Tracts in Mathematics 189 (Cambridge University Press, Cambridge, 2012). · Zbl 1246.47001
[19] J.Li and T.Sahlsten, ‘Fourier transform of self‐affine measures’, Adv. Math.374 (2020) 107349. · Zbl 1448.42012
[20] J.Liouville, Extension au cas des trois dimensions de la question du tracé géographique. Note VI in the Appendix to G. Monge, Application de l’Analyse à la Géometrie, 5th edn (Bachelier, Paris, 1850) 609-616.
[21] V.Mayer and M.Urbański, ‘Finer geometric rigidity of limit sets of conformal IFS’, Proc. Amer. Math. Soc.131 (2003) 3695-3702. · Zbl 1090.37022
[22] C.Moreira and J.Yoccoz, ‘Stable intersections of Cantor sets with large Hausdorff dimension’, Ann. of Math. (2)154 (2001) 45-96. · Zbl 1195.37015
[23] S. E.Newhouse, ‘The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms’, Publ. Math. Inst. Hautes Études Sci.50 (1979) 101-151. · Zbl 0445.58022
[24] K.Nikodem and Z.Páles, ‘Minkowski sums of Cantor‐type sets’, Colloq. Math.119 (2010) 95-108. · Zbl 1194.28011
[25] D.Oberlin and R.Oberlin, ‘Dimensions of sums with self‐similar sets’, Colloq. Math.147 (2017) 43-54. · Zbl 1360.28006
[26] J.Palis and F.Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractor. Cambridge Studies in Advanced Mathematics 35 (Cambridge University Press, Cambridge, 1993). · Zbl 0790.58014
[27] N.Patzschke, ‘Self‐conformal multifractal measures’, Adv. Appl. Math.19 (1997) 486-513. · Zbl 0912.28007
[28] V. Y.Protasov and A. S.Voynov, ‘Matrix semigroups with constant spectral radius’, Linear Algebra Appl.513 (2017) 376-408. · Zbl 1359.15010
[29] Y. G.Reshetnyak, Stability theorems in geometry and analysis, Mathematics and its Applications 304 (Kluwer Academic Publishers Group, Dordrecht, 1994). · Zbl 0925.53005
[30] R. T.Rockafellar, Convex analysis, Princeton Mathematical Series 28 (Princeton University Press, Princeton, NJ, 1970). · Zbl 0202.14303
[31] E.Rossi and P.Shmerkin, ‘On measures that improve \(L^q\) dimension under convolution’, Rev. Mat. Iberoam.36 (2020) 2217-2236. · Zbl 1459.28009
[32] K.Simon and K.Taylor, ‘Dimension and measure of sums of planar sets and curves’, Preprint, 2017, arXiv:1707.01407.
[33] K.Simon and K.Taylor, ‘Interior of sums of planar sets and curves’, Math. Proc. Cambridge Philos. Soc.168 (2020) 119-148. · Zbl 1429.28017
[34] B.Solomyak, ‘On the measure of arithmetic sums of Cantor sets’, Indag. Math.8 (1997) 133-141. · Zbl 0876.28015
[35] Y.Takahashi, ‘Sums of two homogeneous Cantor sets’, Trans. Amer. Math. Soc.372 (2019) 1817-1832. · Zbl 1414.28009
[36] Y.Takahashi, ‘Sums of two self‐similar Cantor sets’, J. Math. Anal. Appl.477 (2019) 613-626. · Zbl 1416.28006
[37] J.Väisälä, Lectures on \(n\)‐dimensional quasiconformal mappings. Lecture Notes in Mathematics 229 (Springer, Berlin-New York, 1971). · Zbl 0221.30031
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