Spectral Properties of Continuum Fibonacci Schrodinger Operators

Speaker: 

May Mei

Institution: 

Denison University

Time: 

Tuesday, May 23, 2017 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

In an award winning 2014 paper, Damanik, Fillman, and Gorodetski rigorously established a framework for investigating Schrodinger operators on the real line whose potentials are generated by ergodic subshifts. In the case of the Fibonacci subshift, they also described the asymptotic behavior in the large energy and small coupling settings when the potential pieces are characteristic functions of intervals of equal length. These estimates relied on explicit formulae and calculations, and thus could not be immediately generalized. In joint work with Fillman, we show that when the potential pieces are square integrable, the Hausdorff dimension of the spectrum tends to one in the large energy and small coupling settings.

On self-similar sets with overlaps and inverse theorems for entropy

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, March 14, 2017 - 1:00pm to 2:00pm

Location: 

RH 440R

We discuss an inverse theorem on the structure of pairs of discrete probability measures which has small amount of growth under convolution, and apply this result to self-similar sets with overlaps to show that if the dimension is less than the generic bound, then there are superexponentially close cylinders at all small enough scales. The results are by M.Hochman. 

Approximation by Algebraic Numbers

Speaker: 

Ryan Broderick

Institution: 

UC Irvine

Time: 

Tuesday, May 10, 2016 - 1:00pm to 2:00pm

Location: 

RH 440R

Dirichlet’s approximation theorem states that for every real number x there exist infinitely many rationals p/q with |x-p/q| < 1/q^2. If x is in the unit interval, then viewing rationals as algebraic numbers of degree 1, q is also the height of its primitive integer polynomial, where height means the maximum of the absolute values of the coefficients. This suggests a more general question: How well can real numbers be approximated by algebraic numbers of degree at most n, relative to their heights? We will discuss Wirsing’s conjecture which proposes an answer to this question and Schmidt and Davenport’s proof of the n = 2 case, as well as some open questions.

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