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A unified approach to mass transference principle and large intersection property (2402.00513v3)

Published 1 Feb 2024 in math.NT and math.MG

Abstract: The mass transference principle, discovered by Beresnevich and Velani [Ann Math (2), 2006], is a landmark result in Diophantine approximation that allows us to obtain the Hausdorff measure theory of $\limsup$ set. Another important tool is the notion of large intersection property, introduced and systematically studied by Falconer [J. Lond. Math. Soc. (2), 1994]. The former mainly focuses on passing between full (Lebesgue) measure and full Hausdorff measure statements, while the latter transfers full Hausdorff content statement to Hausdorff dimension. From this perspective, the proofs of the two results are quite similar but often treated in different ways. In this paper, we establish a general mass transference principle from the viewpoint of Hausdorff content, aiming to provide a unified proof for the aforementioned results. More precisely, this principle allows us to transfer the Hausdorff content bounds of a sequence of open sets $E_n$ to the full Hausdorff measure statement and large intersection property for $\limsup E_n$. One of the advantages of our approach is that the verification of the Hausdorff content bound does not require the construction of Cantor-like subset, resulting in a much simpler proof. As an application, we provide simpler proofs for several mass transference principles.

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References (14)
  1. D. Allen and V. Beresnevich. A mass transference principle for systems of linear forms and its applications. Compos. Math. 154 (2018), 1014–1047.
  2. D. Allen and E. Daviaud. A survey of recent extensions and generalisations of the Mass Transference Principle. Preprint arXiv:2306.15535v1, 2023.
  3. V. Beresnevich and S. Velani. A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164 (2006), 971–992.
  4. Y. Bugeaud. Intersective sets and Diophantine approximation. Michigan Math. J. 52 (2004), 667–682.
  5. K. Falconer. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988), 339–50.
  6. K. Falconer. Sets with large intersection properties. J. Lond. Math. Soc. (2) 49 (1994), 267–280.
  7. P. Erdös. Representations of real numbers as sums and products of Liouville numbers. Michigan Math. J. 9 (1962), 59–60.
  8. S. Eriksson-Bique. A new Hausdorff content for limsup sets. Preprint arXiv:2201.13412v1, 2022.
  9. A. Ghosh and D. Nandi. Diophantine approximation, large intersections and geodesics in negative curvature. Acceped by Proc. Lond. Math. Soc. 2024.
  10. M. Hussain and D. Simmons. A general principle for Hausdorff measure. Proc. Am. Math. Soc. 147 (2019), 3897–3904.
  11. T. Persson. A mass transference principle and sets with large intersections. Real Anal. Exchange 47 (2022), 191–205.
  12. B. Wang and J. Wu. Mass transference principle from rectangles to rectangles in Diophantine approximation. Math. Ann. 381 (2021), 243–317.
  13. B. Wang and J. Wu. Hausdorff dimension of the Cartesian product of lim suplimit-supremum\limsuplim sup sets in Diophantine approximation. Trans. Amer. Math. Soc. 377 (2024), 3727–3748.
  14. B. Wang and G. Zhang. A dynamical dimension transference principle for dynamical Diophantine approximation. Math. Z. 298 (2021), 161–191.
Citations (2)

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