Exact Order of Multiplicative Approximation in Beta Dynamical Systems

• Xiaohui Fu ^[1] ; Xinyun Zhang ^[2]
  1. [1] Huazhong University of Science and Technology

     Huazhong University of Science and Technology

     China

     
  2. [2] Nanchang Hangkong University

     Nanchang Hangkong University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • For a real number β > 1, let Tβ : x → βx (mod 1) denote the β-transformation. In this paper, we study the exact order of multiplicative approximation in beta dynamical systems. For a non-decreasing function ψ : R>0 → R>0, letMβ(ψ) consist of points (x, y) ∈ [0, 1]2 satisfying for any > 0, T n β x · T n β y < β −ψ(n)(1−) for infinitely many n ∈ N, but T n β x · T n β y > β −ψ(n)(1+) for all sufficiently large n ∈ N.

    We establish thatMβ(ψ) has Hausdorff dimension 2+η 1+η , where η = lim inf n→∞ ψ(n) n .

• Referencias bibliográficas
  • 1. Beresnevich, V., Haynes, A., Velani, S.: Multiplicative zero-one laws and metric number theory. Acta Arith. 160(2), 101–114 (2013)
  • 2. Beresnevich, V., Haynes, A., Velani, S.: Sums of reciprocals of fractional parts and multiplicative Diophantine approximation. Mem. Amer....
  • 3. Beresnevich, V., Velani, S.: A note on three problems in metric Diophantine approximation, In: Siddhartha, B., Tarun, D., Anish, G., Riddhi,...
  • 4. Blanchard, F.: β-expansions and symbolic dynamics. Theoret. Comput. Sci. 65(2), 131–141 (1989)
  • 5. Bovey, J.D., Dodson, M.M.: The fractional dimension of sets whose simultaneous rational approximations have errors with a small product....
  • 6. Bugeaud, Y.: A note on inhomogeneous Diophantine approximation. Glasg. Math. J. 45(1), 105–110 (2003)
  • 7. Bugeaud, Y.: Sets of exact approximation order by rational numbers.Math. Ann. 327, 171–190 (2003)
  • 8. Chow, S., Yang, L.: Effective equidistribution for multiplicative Diophantine approximation on lines. Invent. Math. 235(3), 973–1007 (2024)
  • 9. Dirichlet, P.G.L.: Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen,...
  • 10. Einsiedler,M., Katok, A., Lindenstrauss, E.: Invariantmeasures and the set of exceptions to Littlewood conjecture. Ann. Math. 164(2),...
  • 11. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn.Wiley, Hoboken, NJ (2003)
  • 12. Fan, A.H., Wang, B.W.: On the lengths of basic intervals in beta expansions. Nonlinearity 25(5), 1329–1343 (2012)
  • 13. Fang, L.L., Wu, M., Li, B.: Approximation orders of real numbers by β-expansions. Math. Z. 296(1), 13–40 (2020)
  • 14. Gallagher, P.X.: Metric simultaneous diophantine approximation. J. London Math. Soc. 37, 387–390 (1962)
  • 15. Ganotaki, C., Persson, T.:On eventually always hitting points. Monatsh. Math. 196(4), 763–784 (2021)
  • 16. Hill, R., Velani, S.: The ergodic theory of shrinking targets. Invent. Math. 119(1), 175–198 (1995)
  • 17. Hill, R., Velani, S.: Metric diophantine approximation in Julia sets of expanding rational maps. Inst. Hautes Études Sci. Publ. Math....
  • 18. Hill, R., Velani, S.: The shrinking target problems for matrix transformations of tori. J. London Math. Soc. 60(2), 381–398 (1999)
  • 19. Kirsebom,M.,Kunde, P., Persson, T.: Shrinking targets and eventually always hitting points for interval maps. Nonlinearity 33(2), 892–914...
  • 20. Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea Publishing Company, New York (1950)
  • 21. Legendre, A.M.: Essai sur la théorie des nombres. Chez Courcier, Imprimeur-Libraire pour les Mathématiques, Paris (1808)
  • 22. Li, B., Liao, L.M., Velani, S., Zorin, E.: The shrinking target problem for matrix transformations of tori: revisiting the standard problem....
  • 23. Parry, W.: On the β-expansions of real numbers. Acta. Math. Acad. Sci. Hung. 11, 401–416 (1960)
  • 24. Persson, T., Schmeling, J.: Dyadic Diophantine approximation and Katok’s horseshoe approximation. Acta Arith. 132(3), 205–230 (2008)
  • 25. Philipp, W.: Some metrical theorems in number theory. Pac. J. Math. 20(1), 109–127 (1967)
  • 26. Pollington, A., Velani, S.: On a problem in simultaneous Diophantine approximation: littlewood conjecture. Acta Math. 185(2), 287–306...
  • 27. Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477–493 (1957)
  • 28. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill Book Co., New York (1976)
  • 29. Schleischitz, J.:Metric results on sumsets andCartesian products of classes ofDiophantine sets.Results Math. 78(6), 215 (2023)
  • 30. Shen, L.M., Wang, B.W.: Shrinking target problems for β-dynamical system. Sci China Math. 56(1), 91–104 (2013)
  • 31. Spencer, D.C.: The lattice points of tetrahedra. J. Math. Phys. Mass. Inst. Tech. 21, 189–197 (1942)
  • 32. Stolz, O.:Vorlesungen überAllgemeineArithmetik:Nach denNeueren Ansichten, vol. 1.B.G.Teubner Verlag, Leipzig (1885)
  • 33. Tian, C., Peng, L.Q.: Two-dimensional exact Diophantine approximation problem in beta-dynamical systems. Acta Math. Sin. Chin. Ser. 1–24
  • 34. Troubetzkoy, S., Schmeling, J.: Inhomogeneous Diophantine approximations and angular recurrence for billiards in polygons. Math. Sb. 194,...
  • 35. Zhang, X.Y., Zhong, W.M.: Exact Diophantine approximation of real numbers by β-expansions. Discrete Contin. Dyn. Syst. 44(9), 2684–2696...