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Barker sequences and Littlewood’s “two-sided conjectures” on polynomials with \(\pm\)1 coefficients. (English) Zbl 0739.11030
Sémin. Anal. Harmonique, Année 1989/90, 139-151 (1990).
[For the entire collection see Zbl 0707.00018.]
The author begins with a very nice survey on Barker sequences and on the strong and weak versions of the Littlewood two-sided conjecture, quoting all known results on these subjects. Let us recall that a Barker sequence is a finite sequence \(\{a_ 0,a_ 1,\dots,a_ n\}\in\{- 1,+1\}^{n+1}\), such that \(\forall j=1,2,\dots,n,\) \(\sum_{k=0}^{n- j}a_ k a_{k+j}\in\{-1,0,1\}\). Turyn’s conjecture (going back to the 60’s) on Barker sequences reads: there are only a finite number of Barker sequences.
On the other hand the Littlewood two-sided conjecture (1966) asserts: for infinitely many integers \(n\) there exists a polynomial \(P\) of degree \(n\), with all its coefficients equal to \(\pm1\), such that \(A\sqrt{n+1}\leq| P(e^{it})|\leq B\sqrt{n+1}\), \(\forall t\in[0,2\pi[\), where \(A\) and \(B\) are positive absolute constants.
The rather surprising result proved by the author at the end of the paper under review is: at least one of the above two conjectures is true!

11K31 Special sequences
42A05 Trigonometric polynomials, inequalities, extremal problems