Roots of Littlewood polynomials with degree 15

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Littlewood polynomial is a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

all of whose coefficients are +1 or −1. Littlewood's problem asks for bounds on the values of such a polynomial on the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

. The answer to this would yield information about the

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...

of binary sequences. They are named for J. E. Littlewood who studied them in the 1950s.

Definition

A polynomial :

p(x) = \sum_^n a_i x^i \,

is a Littlewood polynomial if all the . Littlewood's problem asks for constants and such that there are infinitely many Littlewood polynomials , of increasing degree satisfying :

c_1 \sqrt \le ,  p_n(z) ,  \le c_2 \sqrt . \,

for all on the unit circle. The Rudin–Shapiro polynomials provide a sequence satisfying the upper bound with . In 2019, an infinite family of Littlewood polynomials satisfying both the upper and lower bound was constructed by Paul Balister,

Béla Bollobás Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul E ...

, Robert Morris,

Julian Sahasrabudhe Julian Sahasrabudhe (born May 8, 1988) is a Canadian mathematician who is an assistant professor of mathematics at the University of Cambridge, in their Department of Pure Mathematics and Mathematical Statistics. His research interests are in e ...

, and Marius Tiba.

References

* * *{{cite journal , last1=Balister , first1=Paul , last2=Bollobás , first2=Béla , last3=Morris , first3=Robert , last4=Sahasrabudhe , first4=Julian , last5=Tiba , first5=Marius , date= 9 November 2020 , title= Flat Littlewood polynomials exist , url= , journal= Annals of Mathematics , volume= 192 , issue= 3 , pages= 977–1004 , doi= 10.4007/annals.2020.192.3.6 , access-date=, arxiv=1907.09464 Polynomials Conjectures