On a problem due to Littlewood concerning polynomials with unimodular coefficients
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On a problem due to Littlewood concerning polynomials with unimodular coefficients Kai-Uwe Schmidt
Received: 13 September 2012 / Published online: 16 March 2013 © Springer Science+Business Media New York 2013
Abstract Littlewood raised the question of how slowly fn 44 − fn 42 (where .r denotes the Lr norm on the unit circle) can grow for a sequence of polynomials fn with unimodular coefficients and increasing degree. The results of this paper are the following. For gn (z) =
n−1
eπik
2 /n
zk
k=0
(gn 44
− gn 42 )/gn 32
the limit of is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials hn (z) =
n−1 n−1
e2πij k/n znj +k
j =0 k=0
the limit of (hn 44 − hn 42 )/hn 32 is shown to be 4/π 2 . No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood’s question. It is an open question as to whether such a sequence of polynomials exists. Keywords Polynomial · Restricted coefficients · Norm · Littlewood problem · Merit factor Mathematics Subject Classification (2010) Primary 42A05 · 11B83 · Secondary 94A55 Communicated by Yang Wang. K.-U. Schmidt () Faculty of Mathematics, Otto-von-Guericke University, Universitätsplatz 2, 39106 Magdeburg, Germany e-mail: [email protected]
458
J Fourier Anal Appl (2013) 19:457–466
1 Introduction For real r ≥ 1, the Lr norm of a polynomial f ∈ C[z] on the unit circle is f r =
1 2π
2π
iθ r f e dθ
1/r .
0
There is sustained interest in the Lr norm of polynomials with restricted coefficients (see, for example, Littlewood [14], Borwein [2], and Erdélyi [5] for surveys on selected problems). Littlewood raised the question of how slowly fn 44 − fn 42 can grow for a sequence of polynomials fn with restricted coefficients and increasing degree. This problem is also of interest in the theory of communications, because f 44 equals the sum of squares of the aperiodic autocorrelations of the sequence formed from the coefficients of f [2, p. 122]; in this context one considers the merit factor f 42 /(f 44 − f 42 ). Much work on Littlewood’s question has been done when the coefficients are −1 or 1; see [8] for recent advances. In the situation where the coefficients are restricted to have unit magnitude, the polynomials gn (z) =
n−1
eπik
2 /n
zk
for integral n ≥ 1
k=0
are of particular interest [11–14].1 These polynomials are also the main ingredient in Kahane’s celebrated semi-probabilistic construction of ultra-flat polynomials [9], which disproves a conjecture due to Erd˝os [6]. Write αn =
gn 44 − gn 42 gn 32
√ (note that f 2 = n for every polynomial f of degree n − 1 with unimodular coefficients). Based on the work in [11] and [12] and calculations carried out by Swinnerton-Dyer, Littlewood concluded in [13] that lim αn =
n→∞
√
2−
2 √ ( 2 − 1) = 1.15051 . . . , π
(1)
but expressed doubt in his own conclusion. He knew that 0.604 ≤ αn ≤ 0.656 for 18 ≤ n ≤ 41
(2)
and noted [13
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