×

Characterization of the algebraic difference of special affine Cantor sets. (English) Zbl 07959972

Summary: We investigate some self-similar Cantor sets \(C(l,r,p)\), which we call S-Cantor sets, generated by numbers \(l,r,p \in \mathbb{N}, l+r< p\). We give a full characterization of the set \(C(l_1, r_1, p)-C(l_2, r_2, p)\) which can take one of the form: the interval \([-1,1]\), a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval. As corollaries we give examples of Cantor sets and Cantorvals, which can be easily described using some positional numeral systems.

MSC:

28A80 Fractals
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
11A67 Other number representations

References:

[1] R. Anisca and C. Chlebovec, On the structure of arithmetic sum of Cantor sets with constant ratios of dissection, Nonlinearity 22 (2009), 2127-2140. · Zbl 1185.28011
[2] R. Anisca, C. Chlebovec and M. Ilie, The structure of arithmetic sums of affine Cantor sets, Real Anal. Exchange 37 (2011/2012), no. 2, 324-332. · Zbl 1270.28006
[3] T. Banakh, A. Bartoszewicz, M. Filipczak and E. Szymonik, Topological and measure properties of some self-similar sets, Topol. Methods Nonlinear Anal. 46 (2015), 1013-1028. · Zbl 1362.28009
[4] D. Damanik, A. Gorodetski and B. Solomyak, Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian, Duke Math. J. 164 (2015), 1603-1640. · Zbl 1358.37117
[5] K. I. Eroglu, On the arithmetic sums of Cantor sets, Nonlinearity 20 (2007), 1145-1161. · Zbl 1129.28007
[6] T. Filipczak and P. Nowakowski, Conditions for the difference set of a central Cantor set to be a Cantorval, Results Math. 78 (2023), art. 166. · Zbl 1528.28002
[7] M. Hall, On the sum and product of continued fractions, Ann. Math. 48 (1947), 966-993. · Zbl 0030.02201
[8] B. Hunt, I. Kan and J. Yorke, Intersection of thick Cantor sets, Trans. Amer. Math. Soc. 339 (1993), 869-888. · Zbl 0783.28006
[9] R. Kenyon, Projecting the one-dimensional Sierpinski gasket, Israel J. Math. 97 (1997), 221-238. · Zbl 0871.28006
[10] R.L. Kraft, What’s the difference between Cantor sets? Amer. Math. Monthly 101 (1994), 640-650. · Zbl 0826.28007
[11] A. Kumar, M. Rani and R. Chugh, New 5-adic Cantor sets and fractal string, SpringerPlus 2 (2013), article no. 654.
[12] P. Mendes and F. Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), 329-343. · Zbl 0839.54027
[13] C.G. Moreira and J.C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. Math. 154 (2001), 45-96. · Zbl 1195.37015
[14] S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci. 50 (1979), 101-151. · Zbl 0445.58022
[15] Z. Nitecki, Cantorvals and subsum sets of null sequences, Amer. Math. Monthly 122 (2015), 862-870. · Zbl 1355.40001
[16] J.E. Nymann and R. Saenz, The topological structure of the set of \(P\)-sums of a sequence, Publ. Math. Debrecen 50 (1997), no. 3-4, 305-316. · Zbl 0880.11013
[17] J.E. Nymann and R. Saenz, The topological structure of the set of \(P\)-sums of a sequence, II, Publ. Math. Debrecen 56 (2000), no. 1-2, 77-85. · Zbl 0960.11003
[18] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, Cambridge, 1993. · Zbl 0790.58014
[19] M. Pourbarat, Topological structure of the sum of two homogeneous Cantor sets, Ergodic Theory Dynam. Systems 43 (2023), no 5, 1712-1736, DOI: 10.1017/etds.2021.156. · Zbl 1547.28022
[20] F. Prus-Wisniowski and F. Tulone, The arithmetic decomposition of central Cantor sets, J. Math. Anal. Appl. 467 (2018), 26-31. · Zbl 1396.28020
[21] M. Repicky, Sets of points of symmetric continuity, Arch. Math. Logic 54 (2015), 803-824. · Zbl 1333.26001
[22] A. Sannami, An example of a regular Cantor set whose difference set is a Cantor set with positive measure, Hokkaido Math. J. 21 (1992), 7-24. · Zbl 0787.58028
[23] B. Solomyak, On the measure of arithmetic sums of Cantor sets, Indag. Math. 8 (1997), 133-141. · Zbl 0876.28015
[24] Y. Takahashi, Quantum and spectral properties of the Labyrinth model, J. Math. Phys. 57 (2016), 063506. · Zbl 1342.82147
[25] Y. Takahashi, Sums of two self-similar Cantor sets, J. Math. Anal. Appl. 477 (2019), 613-626. · Zbl 1416.28006
[26] R. Winkler, The order theoretic structure of the set of P-sums of a sequence, Publ. Math. Debrecen 58 (2001), no. 3, 467-490. · Zbl 1062.11007
[27] Y. Zeng, Self-similar subsets of a class of Cantor sets, J. Math. Anal. Appl. 439 (2016), 57-69. · Zbl 1336.28010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.