Littlewood polynomial

A polynomial all of whose coefficients are $\pm1$. Littlewood's problem asks how large the values of such a polynomial must be on the unit circle in the complex plane. The answer to this would yield information about the autocorrelation of binary sequences. They are named for J. E. Littlewood who studied them in the 1950s.

A polynomial $$ p(x) = \sum_{i=0}^n a_i x^i $$ is a Littlewood polynomial if all the $a_i = \pm 1$. Let $\Vert p \Vert$ denote the supremum of $p(z)$ on the unit circle. Littlewood's problem asks for constants $c_1$ and $c_2$ such that there are infinitely many polynomials $p_n$, of increasing degree $n$, such that $$ c_1 \sqrt{n+1} \le \Vert p_n \Vert \le c_2 \sqrt{n+1} \ . $$ The Rudin-Shapiro polynomials provide a sequence satisfying the upper bound with $c_2 = \sqrt 2$. No sequence is known (as of 2008) which satisfies the lower bound.

References

• Peter Borwein; Computational Excursions in Analysis and Number Theory, ser. CMS Books in Mathematics (2002), pp. 2-5,121-132, Springer-Verlag ISBN 0-387-95444-9
• J.E. Littlewood; Some problems in real and complex analysis, (1968), D.C. Heath