The Littlewood problem and non-harmonic Fourier series

The aim of this note is to prove a lower bound of the L
¹
-norm of non-harmonic trigonometric polynomials of the form\begin{equation*}C\sum\limits_{k = 1}^N {\frac{{\left| {{a_k}} \right|}}{k}} \leq \frac{1}{T}\int_{ - T/2}^{T/2} \mid \sum\limits_{k = 1}^N {{a_k}} {e^{2i\pi {\lambda _k}t}}{\mathbf{d}}t\end{equation*}where T > 1, C = C(T) is an explicit constant that depends on T only, λ
k
are real numbers with |λ
k
− λ
l
| ≥ 1 and a
k
are complex numbers. This extends to the non-harmonic case the Littlewood conjecture as solved by McGehee, Pigno, Smith [6] and Konyagin [5] and was previously obtained by Nazarov [7] with a non-explicit constant C.