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Dynamical Systems and Analysis Seminar - Benjamin Bedert

Dates:9 December 2024
Times:14:00 - 14:00
What is it:Seminar
Organiser:Department of Mathematics
Who is it for:University staff, External researchers, Current University students
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  • Department of Mathematics

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  • In category "Seminar"
  • In group "(Maths) Analysis and dynamical systems"
  • In group "(Maths) Maths seminar series"
  • By Department of Mathematics

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
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Yotam Smilansky

yotam.smilansky@manchester.ac.uk

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