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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.

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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


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


254A, Supplement 3: The Gamma function and the functional equation (optional)

In Notes 2, the Riemann zeta function (and more generally, the Dirichlet -functions ) were extended meromorphically into the region in and to the right of the critical strip. 8,413 more words


254A, Supplement 2: A little bit of complex and Fourier analysis

We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we quickly review the portions of complex analysis that we will be using in this course. 5,281 more words


Real analysis relative to a finite measure space

In the traditional foundations of probability theory, one selects a probability space , and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state , and… 9,931 more words