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Fractional parts of polynomials

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James Maynard
University of Oxford
Thu, 2018-09-06 10:15 - 11:00
MPIM Lecture Hall

Let $f_1$,..., $f_k$ be real polynomials with no constant term and degree at most $d$. We will talk about work in progress showing that there are integers $n$ such that the fractional part of each of the $f_i(n)$ is very small, with the quantitative bound being essentially optimal in the k-aspect. This is based on the interplay between Fourier analysis, Diophantine approximation and the geometry of numbers. In particular, the key idea is to find strong additive structure in Fourier coefficients.

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