Geometric minimality conditions for systems of exponentials in \(L_ p[-\pi,\pi]\) (Q1922307)

Among others, the following minimality condition is proved for the system \(e(\Lambda)=\{e^{i\lambda_kx}\}\) of exponentials, where \(\Lambda=\{\lambda_k\in\mathbb{C}:k\in\mathbb{Z}\}\) and \(\lambda_k\neq\lambda_\ell\) whenever \(k\neq\ell\): Suppose that \(1\leq p\leq 2\), \(\varepsilon>-1/2q\), where \(1/p+1/q=1\), and \(0\leq C<1+2\varepsilon\). If \(\lambda_k\in[k+\varepsilon, k+\varepsilon+C]\) for \(k>0\), \(\lambda_k\in[k-\varepsilon-C, k-\varepsilon]\) for \(k<0\), while \(\lambda_0\) is arbitrary, then the system \(e(\Lambda)\) is minimal in the space \(L_p[-\pi,\pi]\).