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On sums of semibounded Cantor sets. (English) Zbl 07731143

Summary: Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets and growth assumptions on the functions, the sums of such sets contain half-lines. We also give examples to show our criteria are sharp in suitable regimes.

MSC:

28A80 Fractals
35J10 Schrödinger operator, Schrödinger equation
52C23 Quasicrystals and aperiodic tilings in discrete geometry
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[1] S. Astels, “Cantor sets and numbers with restricted partial quotients”, Trans. Amer. Math. Soc. 352:1 (2000), 133-170. · Zbl 0967.11026 · doi:10.1090/S0002-9947-99-02272-2
[2] M. Baake and U. Grimm, Aperiodic order, I: A mathematical invitation, Encyclopedia of Mathematics and its Applications 149, Cambridge Univ. Press, 2013. · Zbl 1295.37001 · doi:10.1017/CBO9781139025256
[3] M. Baake and R. V. Moody (editors), Directions in mathematical quasicrystals, CRM Monograph Series 13, Amer. Math. Soc., Providence, RI, 2000. · Zbl 0955.00025 · doi:10.1090/crmm/013
[4] J. Bellissard, “Spectral properties of Schrödinger’s operator with a Thue-Morse potential”, pp. 140-150 in Number theory and physics (Les Houches, 1989), edited by J. M. Luck et al., Springer Proc. Phys. 47, Springer, 1990. · Zbl 0719.58038 · doi:10.1007/978-3-642-75405-0_13
[5] J. Bellissard, A. Bovier, and J.-M. Ghez, “Spectral properties of a tight binding Hamiltonian with period doubling potential”, Comm. Math. Phys. 135:2 (1991), 379-399. · Zbl 0726.58038
[6] J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, “Spectral properties of one-dimensional quasi-crystals”, Comm. Math. Phys. 125:3 (1989), 527-543. · Zbl 0825.58010
[7] D. Damanik, M. Embree, J. Fillman, and M. Mei, “Discontinuities of the integrated density of states for Laplacians associated with Penrose and Ammann-Beenker tilings”, Exp. Math. (online publication May 2023). · doi:10.1080/10586458.2023.2206589
[8] D. Damanik, J. Fillman, and A. Gorodetski, “Continuum Schrödinger operators associated with aperiodic subshifts”, Ann. Henri Poincaré 15:6 (2014), 1123-1144. · Zbl 1292.81052 · doi:10.1007/s00023-013-0264-6
[9] D. Damanik, J. Fillman, and A. Gorodetski, “Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set”, J. Funct. Anal. 280:7 (2021), art. id. 108911. · Zbl 1460.81026 · doi:10.1016/j.jfa.2020.108911
[10] D. Damanik and A. Gorodetski, “Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian”, Comm. Math. Phys. 305:1 (2011), 221-277. · Zbl 1232.81016 · doi:10.1007/s00220-011-1220-2
[11] D. Damanik, A. Gorodetski, and W. Yessen, “The Fibonacci Hamiltonian”, Invent. Math. 206:3 (2016), 629-692. · Zbl 1359.81108 · doi:10.1007/s00222-016-0660-x
[12] D. Damanik and D. Lenz, “A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem”, Duke Math. J. 133:1 (2006), 95-123. · Zbl 1118.37009 · doi:10.1215/S0012-7094-06-13314-8
[13] J. Fillman, Y. Takahashi, and W. Yessen, “Mixed spectral regimes for square Fibonacci Hamiltonians”, J. Fractal Geom. 3:4 (2016), 377-405. · Zbl 1370.47033 · doi:10.4171/JFG/40
[14] T. Fujiwara, M. Arai, T. Tokihiro, and M. Kohmoto, “Localized states and self-similar states of electrons on a two-dimensional Penrose lattice”, Phys. Rev. B (3) 37:6 (1988), 2797-2804. · doi:10.1103/PhysRevB.37.2797
[15] B. Helffer and A. Mohamed, “Asymptotic of the density of states for the Schrödinger operator with periodic electric potential”, Duke Math. J. 92:1 (1998), 1-60. · Zbl 0951.35104 · doi:10.1215/S0012-7094-98-09201-8
[16] A. Hof, “Some remarks on discrete aperiodic Schrödinger operators”, J. Statist. Phys. 72:5-6 (1993), 1353-1374. · Zbl 1101.39301 · doi:10.1007/BF01048190
[17] A. Hof, “A remark on Schrödinger operators on aperiodic tilings”, J. Statist. Phys. 81:3-4 (1995), 851-855. · Zbl 1081.82521 · doi:10.1007/BF02179262
[18] Y. E. Karpeshina, Perturbation theory for the Schrödinger operator with a periodic potential, Lecture Notes in Mathematics 1663, Springer, 1997. · Zbl 0883.35002 · doi:10.1007/BFb0094264
[19] S. Klassert, D. Lenz, and P. Stollmann, “Discontinuities of the integrated density of states for random operators on Delone sets”, Comm. Math. Phys. 241:2-3 (2003), 235-243. · Zbl 1098.82016 · doi:10.1007/s00220-003-0920-7
[20] M. Kohmoto and B. Sutherland, “Electronic states on a Penrose Lattice”, Phys. Rev. Lett. 56:25 (1986), 2740-2743. · doi:10.1103/PhysRevLett.56.2740
[21] D. Lenz, C. Seifert, and P. Stollmann, “Zero measure Cantor spectra for continuum one-dimensional quasicrystals”, J. Differential Equations 256:6 (2014), 1905-1926. · Zbl 1351.47030 · doi:10.1016/j.jde.2013.12.003
[22] D. Lenz and P. Stollmann, “Delone dynamical systems and associated random operators”, pp. 267-285 in Operator algebras and mathematical physics (Constant,a, 2001), edited by J.-M. Combes et al., Theta, Bucharest, 2003. · Zbl 1285.47044
[23] D. Lenz and P. Stollmann, “An ergodic theorem for Delone dynamical systems and existence of the integrated density of states”, J. Anal. Math. 97 (2005), 1-24. · Zbl 1131.37015 · doi:10.1007/BF02807400
[24] Q. Liu and Y. Qu, “On the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian”, Comm. Math. Phys. 338:2 (2015), 867-891. · Zbl 1333.47026 · doi:10.1007/s00220-015-2377-x
[25] Q. Liu, Y. Qu, and X. Yao, “Unbounded trace orbits of Thue-Morse Hamiltonian”, J. Stat. Phys. 166:6 (2017), 1509-1557. · Zbl 1376.82038 · doi:10.1007/s10955-017-1726-x
[26] A. McDonald and K. Taylor, “Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse gap lemma”, preprint, 2021. · Zbl 07725804
[27] M. Mei, “Spectra of discrete Schrödinger operators with primitive invertible substitution potentials”, J. Math. Phys. 55:8 (2014), 082701, 22. · Zbl 1302.82112 · doi:10.1063/1.4886535
[28] M. Mei and W. Yessen, “Tridiagonal substitution Hamiltonians”, Math. Model. Nat. Phenom. 9:5 (2014), 204-238. · Zbl 1309.47031 · doi:10.1051/mmnp/20149514
[29] S. E. Newhouse, “Nondensity of axiom \[\text{A}(\text{a})\] on \[S^2 \]”, pp. 191-202 in Global analysis (Berkeley, Calif., 1968), edited by S.-s. Chern and S. Smale, Proc. Sympos. Pure Math. 14-16, Amer. Math. Soc., Providence, R.I., 1970. · Zbl 0206.25801
[30] S. E. Newhouse, “The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms”, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101-151. · Zbl 0445.58022
[31] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractors, Cambridge Studies in Advanced Mathematics 35, Cambridge Univ. Press, 1993. · Zbl 0790.58014
[32] L. Parnovski, “Bethe-Sommerfeld conjecture”, Ann. Henri Poincaré 9:3 (2008), 457-508. · Zbl 1201.81054 · doi:10.1007/s00023-008-0364-x
[33] V. N. Popov and M. M. Skriganov, “Remark on the structure of the spectrum of a two-dimensional Schrödinger operator with periodic potential”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 109 (1981), 131-133. In Russian; translated in J. Soviet Math. 24:2 (1984), 239-240. · Zbl 0532.47035 · doi:10.1007/BF01087244
[34] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry”, Phys. Rev. Lett. 53 (1984), 1951-1953. · doi:10.1103/PhysRevLett.53.1951
[35] K. Simon and K. Taylor, “Interior of sums of planar sets and curves”, Math. Proc. Cambridge Philos. Soc. 168:1 (2020), 119-148. · Zbl 1429.28017 · doi:10.1017/s0305004118000580
[36] M. M. Skriganov, “Proof of the Bethe-Sommerfeld conjecture in dimension \[2\]”, Dokl. Akad. Nauk SSSR 248:1 (1979), 39-42. In Russian; translated in Soviet Math. Dokl. 20 (1979), 956-959. · Zbl 0435.35028
[37] M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators”, Trudy Mat. Inst. Steklov. 171 (1985), 3-122. In Russian; translated in Proc. Steklov Inst. Math. 171 (1987), 1-121. · Zbl 0615.47004
[38] M. M. Skriganov, “The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential”, Invent. Math. 80:1 (1985), 107-121. · Zbl 0578.47003 · doi:10.1007/BF01388550
[39] A. Sütő, “The spectrum of a quasiperiodic Schrödinger operator”, Comm. Math. Phys. 111:3 (1987), 409-415. · Zbl 0624.34017
[40] A. Sütő, “Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian”, J. Statist. Phys. 56:3-4 (1989), 525-531. · Zbl 0712.58046 · doi:10.1007/BF01044450
[41] Y. Takahashi, “Quantum and spectral properties of the labyrinth model”, J. Math. Phys. 57:6 (2016), art. id. 063506. · Zbl 1342.82147 · doi:10.1063/1.4953379
[42] Y. Takahashi, “Products of two Cantor sets”, Nonlinearity 30:5 (2017), 2114-2137. · Zbl 1366.28005 · doi:10.1088/1361-6544/aa6761
[43] O. A. Veliev, “The spectrum of multidimensional periodic operators”, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 49 (1988), 17-34. In Russian; translated in J. Soviet Math. 49:4 (1990), 1045-1058. · Zbl 0696.47003 · doi:10.1007/BF02216095
[44] A. Yavicoli, “Patterns in thick compact sets”, Israel J. Math. 244:1 (2021), 95-126. · Zbl 1483.28012 · doi:10.1007/s11856-021-2173-6
[45] A. Yavicoli, “Thickness and a gap lemma in \[\mathbb{R}^d\]”, preprint, 2022. · Zbl 1483.28012
[46] H. Yu, “Fractal projections with an application in number theory”, Ergodic Theory Dynam. Systems 43:5 (2023), 1760-1784 · Zbl 1527.11008 · doi:10.1017/etds.2022.2
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