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254B, Notes 7: Sieving and expanders

In this final set of course notes, we discuss how (a generalisation of) the expansion results obtained in the preceding notes can be used for some nnumber-theoretic applications, and in particular to locate almost primes inside orbits of thin groups, following the work of Bourgain, Gamburd, and Sarnak. We will not attempt here to obtain [...]
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Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further

Academics / Mathematics : Terence Tao's Blog (last year)

This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on ... Read Post

254B, Notes 4: The Bourgain-Gamburd expansion machine

Academics / Mathematics : Terence Tao's Blog (3 years ago)

We have now seen two ways to construct expander Cayley graphs. The first, discussed in Notes 2, is to use Cayley graphs that are projections of an infinite Cayley graph on a group with Kazhdan’s property (T). The second, discussed i... Read Post

254B, Lecture Notes 7: The transference principle, and linear equations in primes

Academics / Mathematics : Terence Tao's Blog (5 years ago)

In this, the final lecture notes of this course, we give discuss one of the motivating applications of the theory developed thus far, namely to count solutions to linear equations in primes (or in dense subsets of primes ). Unfortun... Read Post

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