Substitutions and bounded remainder sets.

*(Russian. English summary)*Zbl 1445.11076One can begin with the author’s abstract:

“The paper is devoted to the multidimensional problem of distribution of fractional parts of a linear function. A subset of a multidimensional torus is called a bounded remainder set if the remainder term of the multidimensional problem of the distribution of the fractional parts of a linear function on this set is bounded by an absolute constant. We are interested not only in the individual bounded remainder sets but also in toric tilings into such sets.

A new class of tilings of a d-dimensional torus into sets of \((d+1)\) types is introduced. These tilings are defined in combinatorics and geometric terms and are called generalized exchanged tilings. It is proved that all generalized exchanged toric tilings consist of bounded remainder sets. Corresponding estimate of the remainder term is effective. We also find conditions that ensure that the estimate of the remainder term for the sequence of generalized exchanged toric tilings does not depend on the concrete tiling in the sequence.

Using the Arnoux-Ito theory of geometric substitutions we introduce a new class of generalized exchanged tilings of multidimensional tori into bounded remainder sets with an effective estimate of the remainder term. Earlier similar results were obtained in the twodimensional case for one specific substitution. This is a geometric version of well-known Rauzy substitution. With the help of the passage to the limit, another class of generalized exchanged toric tilings into bounded remainder sets with fractal boundaries is constructed (so-called generalized Rauzy fractals).”

A survey on the problem on constructing tilings of multidimensional tori into sets of bounded remainder is given, the interest to such tilings is justified.

This paper consists of the following items:

– Exchanged tilings.

– Geometric substitutions.

– Geometric substitutions and generalized exchanged tilings.

– The Rauzy fractals.

The author notes that the results proven in this paper are generalizations of certain known results. Relations between some results of this research and known results are discussed. Some open problems are described.

“The paper is devoted to the multidimensional problem of distribution of fractional parts of a linear function. A subset of a multidimensional torus is called a bounded remainder set if the remainder term of the multidimensional problem of the distribution of the fractional parts of a linear function on this set is bounded by an absolute constant. We are interested not only in the individual bounded remainder sets but also in toric tilings into such sets.

A new class of tilings of a d-dimensional torus into sets of \((d+1)\) types is introduced. These tilings are defined in combinatorics and geometric terms and are called generalized exchanged tilings. It is proved that all generalized exchanged toric tilings consist of bounded remainder sets. Corresponding estimate of the remainder term is effective. We also find conditions that ensure that the estimate of the remainder term for the sequence of generalized exchanged toric tilings does not depend on the concrete tiling in the sequence.

Using the Arnoux-Ito theory of geometric substitutions we introduce a new class of generalized exchanged tilings of multidimensional tori into bounded remainder sets with an effective estimate of the remainder term. Earlier similar results were obtained in the twodimensional case for one specific substitution. This is a geometric version of well-known Rauzy substitution. With the help of the passage to the limit, another class of generalized exchanged toric tilings into bounded remainder sets with fractal boundaries is constructed (so-called generalized Rauzy fractals).”

A survey on the problem on constructing tilings of multidimensional tori into sets of bounded remainder is given, the interest to such tilings is justified.

This paper consists of the following items:

– Exchanged tilings.

– Geometric substitutions.

– Geometric substitutions and generalized exchanged tilings.

– The Rauzy fractals.

The author notes that the results proven in this paper are generalizations of certain known results. Relations between some results of this research and known results are discussed. Some open problems are described.

Reviewer: Symon Serbenyuk (Kyiv)

##### MSC:

11K06 | General theory of distribution modulo \(1\) |

11K38 | Irregularities of distribution, discrepancy |

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |

##### Keywords:

uniform distribution; bounded remainder sets; toric tilings; unimodular Pisot substitutions; geometric substitutions##### References:

[1] | E. Hecke, “Eber Analytische Funktionen und die Verteilung van Zahlen mod Eins”, Math. Sem. Hamburg Univ., 5 (1921), 54-76 · JFM 48.0184.02 |

[2] | P. Erdös, “Problems and results on diophantine approximations”, Compositio Math., 16 (1964), 52-65 · Zbl 0131.04803 |

[3] | H. Furstenberg, M. Keynes, L. Shapiro, “Prime flows in topological dynamics”, Israel J. Math., 14:1 (1973), 26-38 · Zbl 0264.54030 |

[4] | S. Grepstad, N. Lev, “Sets of bounded discrepancy for multi-dimensional irrational rotation”, Geometric and Functional Analysis, 25:1 (2015), 87-133 · Zbl 1318.11097 |

[5] | A. Heynes, H. Koivusalo, “Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices”, Israel J. Math., 212:1 (2016), 189-201 · Zbl 1341.11043 |

[6] | M. Kelly, L. Sadun, “Patterns equivariant cohomology and theorems of Kesten and Oren”, Bull. London Math. Soc., 47:1 (2015), 13-20 · Zbl 1402.52023 |

[7] | H. Kesten, “On a conjecture of Erdös and Szüsz related to uniform distribution mod 1”, Acta Arithmetica, 12:2 (1966), 193-212 · Zbl 0144.28902 |

[8] | P. Liardet, “Regularities of distribution”, Compositio Math., 61:3 (1987), 267-293 · Zbl 0619.10053 |

[9] | R. Szüsz, “Uber die Verteilung der Vielfachen einer Komplexen Zahl nach dem Modul des Einheitsquadrats”, Acta Math. Acad. Sci. Hungar., 5:1-2 (1954), 35-39 · Zbl 0058.03503 |

[10] | G. Rauzy, “Nombres algébriques et substitutions”, Bull. Soc. Math. France, 110 (1982), 147-178 · Zbl 0522.10032 |

[11] | V. Baladi, D. Rockmore, N. Tongring, C. Tresser, “Renormalization on the \(n\)-dimensional torus”, Nonlinearity, 5:5 (1992), 1111-1136 · Zbl 0761.58008 |

[12] | A. V. Shutov, “The two-dimensional Hecke-Kesten problem”, Chebyshevskii Sbornik, 12:2 (2011), 151-162 · Zbl 1306.11055 |

[13] | V. G. Zhuravlev, “Exchanged toric developments and bounded remainder sets”, Journal of Mathematical Sciences, 184:6 (2012), 716-745 · Zbl 1272.11087 |

[14] | V. G. Zhuravlev, “Multidimensional Hecke theorem on the distribution of fractional parts”, St. Petersburg Mathematical Journal, 24:1 (2013), 71-97 · Zbl 1273.11121 |

[15] | V. Berthe, S. Ferenczi, L. Q. Zamboni, “Interactions between dynamics, arithmetics, and combinatorics: the good, the bad, and the ugly”, Algebraic and Topological Dynamics, 385 (2005), 333-364 · Zbl 1156.37301 |

[16] | N. Chevallier, “Coding of a translation of the two-dimensional torus”, Monatshefte für Mathematik, 157:2 (2009), 101-130 · Zbl 1171.37009 |

[17] | A. V. Shutov, “Multidimensional generalization of sums of fraction parts and their applications to number theory”, Chebyshevskii Sbornik, 14:1 (2013), 104-118 · Zbl 1434.11148 |

[18] | A. V. Shutov, “Trigonometric sums over one-dimensional quasilattices of arbitrary codimension”, Mathematical notes, 97:5-6 (2015), 791-802 · Zbl 1331.11062 |

[19] | V. Berthe, “Arithmetic discrete planes are quasicrystals”, DGCI 2009, 15th IAPR International Conference on Discrete Geometry for Computer Imagery, Springer, 2009, 1-12 · Zbl 1261.52013 |

[20] | A. V. Shutov, A. V. Maleev, “Generalized Rauzy fractals and quasiperiodic tilings”, Classification and Application of Fractals: New Reserch, Nova Publishers, 2012, 55-111 |

[21] | A. V. Shutov, A. V. Maleev, “Quasiperiodic plane tilings based on stepped surfaces”, Acta Crystallographica, A64 (2008), 376-382 · Zbl 1370.52066 |

[22] | A. V. Shutov, A. V. Maleev, V. G. Zhuravlev, “Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry”, Acta Crystallographica, A66 (2010), 427-437 |

[23] | A. A. Abrosimova, “BR-sets”, Chebyshevskii Sbornik, 16:2 (2015), 8-22 · Zbl 1436.11095 |

[24] | A. A. Shutov, A. V. Zhukova, “On the distribution function of the remainder term on bounded remainder sets”, Chebyshevskii Sbornik, 17:1 (2016), 90-107 · Zbl 1440.11128 |

[25] | V. G. Zhuravlev, “Rauzy tilings and bounded remainder sets on the torus”, Journal of Mathematical Sciences, 137:2 (2006), 4658-4672 · Zbl 1158.11331 |

[26] | Kuznetsova D. V., Shutov A. V., “Exchanged Toric Tilings, Rauzy Substitution, and Bounded Remainder Sets”, Mathematical Notes, 98:5-6 (2015), 932-948 · Zbl 1387.11052 |

[27] | N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Springer, 2001 · Zbl 1014.11015 |

[28] | P. Arnoux, S. Ito, “Pisot substitutions and Rauzy fractals”, Bull. Belg. Math. Soc. Simon Stevin, 8:2 (2001), 181-207 · Zbl 1007.37001 |

[29] | A. V. Shutov, “Exchanged toric tilings and the multidimensional Hecke-Kesten problem”, Uchenye zapiski Orlovskogo gosudarstvennogo universiteta, 6:2 (2012), 249-253 |

[30] | Kuznetsova D. V., Shutov A. V., “On the first return times for toric rotations”, Chebyshevskii Sbornik, 14:4 (2013), 127-130 · Zbl 1425.37008 |

[31] | Zhuravlev V. G., “One-dimensional Fibonacci tilings”, Izvestiya: Mathematics, 71:2 (2007), 307-340 · Zbl 1168.11006 |

[32] | A. V. Shutov, Generalized Fibonacci tilings and their applications, Lambert Academic Publishing, 2012, 144 pp. |

[33] | S. Ito, H. Rao, “Atomic surfaces, tilings and coincidences I. Irreducible case”, Israel J. Math., 153:1 (2006), 129-156 · Zbl 1143.37013 |

[34] | M. Barge, J. Kwapisz, “Geometric theory of unimodular Pisot substitutions”, Amer. J. Math., 128:5 (2006), 1219-1282 · Zbl 1152.37011 |

[35] | V. Berthe, A. Siegel, J. Thuswaldner, “Substitutions, Rauzy fractals, and tilings”, Combinatorics, Automata and Number Theory, Cambridge University Press, 2010, 248-323 · Zbl 1247.37015 |

[36] | S. Akiyama, M. Barge, V. Berthe, J. Y. Lee, A. Siegel, “On the Pisot substitution conjecture”, Mathematics of Aperiodic Order, Springer, 2015, 33-72 · Zbl 1376.37043 |

[37] | V. Berthe, J. Bourdon, T. Jolivet, A. Siegel, “A combinatorial approach to products of Pisot substitutions”, Ergodic Theory Dynam. Systems, 36:6 (2016), 1757-1794 · Zbl 1378.37036 |

[38] | V. G. Zhuravlev, “Induced bounded remainder sets”, St. Petersburg Mathematical Journal, 28:5 (2017), 671-688 · Zbl 1372.52024 |

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