Littlewood polynomials with smallL4norm

Abstract

Littlewood asked how small the ratio ||f || 4 /||f || 2 (where ||·|| α denotes the L α norm on the unit circle) can be for polynomials f having all coefficients in {1, −1}, as the degree tends to infinity. Since 1988, the least known asymptotic value of this ratio has been 4 7/6, which was conjectured to be minimum. We disprove this conjecture by showing that there is a sequence of such polynomials, derived from the Fekete polynomials, for which the limit of this ratio is less than 4 22/19.

Low Correlation Sequences From Linear Combinations of Characters

“…. that has already been obtained in [14,19,13] with the modified Legendre sequences h s,ℓ p . In their paper, they do not discuss crosscorrelation, which is the main concern here, in view of the discussion in the previous paragraph.…”

“…This is achieved by sequences derived from finite field characters, specifically quadratic characters (also known as Legendre symbols). See [14,Theorem 1.1], [19,Theorem 1.5], and [13] for details. Other sequences with good correlation properties derived from finite field characters include the maximal linear recursive sequences (msequences), which are used extensively in radar and communications networks.…”

Aperiodic Crosscorrelation of Sequences Derived From Characters

“…Note that if χ : F * p → C is a multiplicative character, then the conjugate character χ indicates the multiplicative character with χ(a) = χ(a) = χ(a) −1 for all a ∈ F * p . Also recall the definition of the function Ω from (9). As with autocorrelation, there is a marked difference in behavior between the quadratic character (Legendre symbol) and all other nontrivial multiplicative characters.…”

“…Recall the function Ω defined in (9). Using the cardinalities of A, B, C, and D computed in Lemmata 7.3-7.5 below, we obtain f g 2…”

“…Proof. We write 2Z + 1 for the set of all odd integers, and then the definition (9) of Ω shows us that Ψ(x) = n∈2Z+1 max(0, 1 − |n|/(2x)) 2 . So if x ∈ (0, 1/2], we see that Ψ(x) = 0.…”