One Day Ergodic Theory Meeting at Warwick
Speakers
Jungwon Lee
Péter Varjú
Giovanni Forni
One Day Ergodic Theory Meeting at Warwick Schedule
Wednesday February 15th, Zeeman Building, MS.04
2:00 - 3:00 PM
Péter Varjú
Title: Diophantine approximation in missing digit sets and Fourier l^1 dimension
Abstract: A missing digit set is subset of [0,1] that can be defined by restricting the digits that can occur in the expansion of the elements in some base. An example is the middle third Cantor set. I will discuss the question of how big the set of those numbers is in a
missing digit set that can be approximated by rationals p/q with error q^-a for some positive real a. The problem comes in two flavours, we can either look at approximation by all rationals in [0,1] or we can
restrict to those contained in the missing digit sets.
This circle of problems is related to a quantity called Fourier l^1 dimension introduced by Han Yu. I will explain this connection and also talk about how Fourier l^1 dimension can be approximated.
Work in progress, joint with Demi Allen, Sam Chow and Han Yu.
3:00-3:30 PM
Tea Break
3:30-4:30 PM
Jungwon Lee
Title: Euclidean algorithms are Gaussian over imaginary quadratic fields.
Abstract: Baladi and Vallée exhibited the limit Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers with bounded denominators based on transfer operator analysis. We give an analogue of the result for complex continued fractions over imaginary quadratic number fields and discuss applications in the theory of L-functions of GL2 (joint with Dohyeong Kim and Seonhee Lim).
4:45-5:45 PM
Giovanni Forni
Title: Scaling in parabolic dynamics and number theory
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Abstract: We will outline a method based on scaling to prove quantitative (unique) ergodicity for parabolic homogeneous flows and give two applications:
the first, to``twisted horocycle flows'' leads to bounds on Fourier coefficients of cusp forms; the second, to a class of higher step nilflows, to bounds on Weyl sums
of higher degree polynomials. We recover results in analytic number theory by Good, Bernstein-Reznikov (for the first example), Wooley and Bourgain, Demeter,
Guth (for the second example). This is joint work with L. Flaminio and J. Tanis (GAFA 2016) on the twisted horocycle flows and L. Flaminio on nilflows.
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6:15 PM