Convergence near the boundary

Analysis

DiscApproxId

In the above pdf, I consider in a unified manner the question of convergence of a power series on the boundary of their radius of convergence, and of Dirichlet series on the boundary of their abscissa of convergence. I end with an application to L(1,\chi), where \chi is the nontrivial multiplicative character modulo 4.

This was motivated by these notes of Keith Conrad and a discussion with Kyle Pratt.

A genus calculation for quotients of upper half plane

Riemann Surfaces,

GenusCalculations

In the above pdf, we calculate the genus of some Riemann surfaces, quotients of the upper half plane by \Gamma_0(p), where p is some prime.

The idea is to project the Riemann surface to \hat{\mathbb{C}} and to apply the Riemann Hurwitz formula to this map.  I include a picture that allows one to visualize the this projection map, which is basically just an appropriate folding of parts of the complex plane.

This post arose after making these calculations one afternoon with Ravi Donepudi, a fellow number theory student here at UIUC. Also, I would like to thank Junxian Li (also in UIUC number theory) for providing the picture and useful discussions.

Gauss Sums and Hausdorff-Young

Harmonic Analysis, Number Theory, ,

GaussSums-Hausdorff Young

Let 1 \leq p \leq 2. The classical Hausdorff-Young inequality asserts that for any f \in L^p(\mathbb{T}), there is a A_p (equal to 1 in the present case) such that ||\hat{f}||_{p/(p-1)} \leq A_p||f||_p. There are examples that show one cannot allow p > 2. In the above pdf, we provide a construction, motivated from number theory, that shows p > 2 is impossible in the Hausdorff-Young inequality. One can think of this as a pseudo-random approach, as opposed to the random approach employed by, for instance, an application of Khintchine’s inequality.

We will first prove a discrete analog. For those a bit rusty, you are invited to check out my notes on the discrete Fourier transform.

I’d like to thank to of my fellow graduate students, Derek Jung and Xiao Li, for pointing out some mistakes in a previous version. I would also like to thank Sergei Konyagin for making me aware of the Shapiro-Rudin polynomials.

Lastly, I have not seen this example in the literature. If you have seen it, please let me know.

Counting lattice points

Number Theory

LatticePoints

The above pdf contains some notes I wrote up on using (for the most part) elementary methods to count lattice points in dilates of the unit ball of \ell_p(\mathbb{R}^d). There are some connections to the Dirichlet kernel, Waring’s problem, smooth number estimates, and probability.

Motivated by a nice talk in the graduate student analysis seminar provided by Hadrian Quan, I make a quick connection to a particular case of Weyl’s law.

Uniform boundedness and Fourier series

Functional Analysis, Harmonic Analysis

UniformBoundedness

The above pdf contains some notes on the uniform boundedness principle of functional analysis. The majority of notes is from the perspective of proving the existence of a continuous function whose Fourier series diverges at a single point. By seeing the concepts in the same light, my goal was to gain some intuition for both.

I’d like to thank two of my fellow graduate students, Chris Gartland and Hadrian Quan, for their useful suggestions.

My fellow graduate student, Martino Fassina, showed me this ridiculously simple proof of uniform boundedness.