Dynamical Systems and Analysis Seminar - Benjamin Bedert
| Dates: | 9 December 2024 |
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| Times: | 14:00 - 14:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Benjamin Bedert (University of Oxford)
Title: Zeros of polynomials with restricted coefficients: a problem of Littlewood
Abstract: The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $0,2\pi$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound.
A closely related question due to Borwein, Erdelyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.
Room: Frank Adams 1
Further information: https://personalpages.manchester.ac.uk/staff/yotam.smilansky/dynamics_analysis
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Frank Adams 1
Alan Turing Building
Manchester
