Diophantine Approximation through the Mass Transference Principle

DATES: March 21-24, 2022 (5 sessions)
TIME: 11:00 - 12:30. On Wednesday it will ends at 12:00 (a total of 7 hours)
LOCATION: BCAM and Online

ABSTRACT:

The goal of this course is to prove Jarnak´s theorem. In 1931 Jarnak studied the size of
the set

W(r) := {x ∈ [0,1] | |x - p/q| ≦ 1/pr for infinitely many fractions p/q},

where r ≧ 2, and concluded that its Hausdorff dimension is 2/r and that its Hausdorff measure at 2/r is infinite. In 2006 Beresnevich and Velani introduced the Mass Transference Principle, a powerful method to compute the dimension of lim sup-sets, so, in particular, the dimension of W(r). We will also see some applications.

CONTENT:
Diophantine approximation, history and motivation.
Basic concepts: Hausdorff dimension and properties; Frostman measures; and some general principles.
Statement and proof of the Mass Transference Principle.
How to deduce Jarnak´s theorem from the Mass Transference Principle and other applications.
If time permits, recent extensions of the Mass Transference Principle.

REFERENCES:
[1] V. Beresnevich, D. Dickinson, and S. Velani. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc., 179(846):x+91, 2006.
[2] V. Beresnevich and S. Velani. A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2), 164(3):971-992, 2006.
[3] K. Falconer. Fractal geometry. John Wiley & Sons, Inc., Hoboken, NJ, second edition, 2003. Mathematical foundations and applications.
[4] V. Jarnak. Öber die simultanen diophantischen Approximationen. Math. Z., 33(1):505-543, 1931.
[5] B. Wang and J. Wu. Mass transference principle from rectangles to rectangles in diophantine approximation. Preprint arXiv:1909.00924v2 [math.NT].


*Registration is free, but mandatory before March 14th. To sign-up go to https://forms.gle/FTb8GKtUjCyzwzmbA and fill the registration form.

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