Extremal Mahler measures and \(L_s\) norms of polynomials related to Barker sequences (Q2839349)

Recall that a sequence \(a_0, a_1,\dots, a_n \in \{-1,1\}\) satisfying \(\sum_{j=0}^{n-k} a_j a_{k+j} \in \{-1,0,1\}\) for each \(k=0,1,\dots,n\) is called a \textit{Barker} sequence and a corresponding polynomial \(\sum_{j=0}^n a_j z^j\) a \textit{Barker polynomial}. The results of these paper are related to the (still open) conjecture asserting that there are only finitely many Barker sequences. In particular, the authors define \(\mathcal{L}P_n\) to be the class of Laurent polynomials \(P\) of the form NEWLINE\[NEWLINEP(z)=n+1+\sum_{k=1, > k -\text{odd}}^{n} c_k (z^k+z^{-k}),NEWLINE\]NEWLINE where \(c_k \in \{-1,1\}\). (These arise in a natural way in connection with Barker polynomials of odd degree.) Let \(R_n\) be the polynomial from the class \(\mathcal{L}P_n\) with all coefficients \(c_k=1\). The authors prove that \(M(P) \geq M(R_n)\) for each \(P \in \mathcal{L}P_n\) which implies that for any Barker polynomial \(p\) of degree \(n\) we must have NEWLINE\[NEWLINEM(p)^2 \geq n-\frac{2}{\pi} \log n +O(1).NEWLINE\]