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Equidistribution for multidimensional polynomial phases

The equidistribution theorem asserts that if is an irrational phase, then the sequence is equidistributed on the unit circle, or equivalently that

for any continuous (or equivalently, for any smooth) function . 2,002 more words

Expository

A differentiation identity

Here’s a cute identity I discovered by accident recently. Observe that

and so one can conjecture that one has

when is even, and

when is odd. 131 more words

Math.CA

Cancellation for the multilinear Hilbert transform

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from… 417 more words

Math.CO

A remark on the lonely runner conjecture

The lonely runner conjecture is the following open problem:

Conjecture 1 Suppose one has runners on the unit circle , all starting at the origin and moving at different speeds.

1,296 more words
Expository

The ergodic theorem and Gowers-Host-Kra seminorms without separability or amenability

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has… 1,351 more words

Expository

254A, Notes 5: Bounding exponential sums and the zeta function

We return to the study of the Riemann zeta function , focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of… 4,209 more words

Math.CA