Weighted Paley-Wiener spaces sharing a majorant-weight

We point out a need to slightly modify the statement of Theorem 2 in Yurii Lyubarskii and Kristian Seip's work Weighted Paley-Wiener spaces. This last theorem lists all weighted Paley-Wiener spaces (in reduced form) sharing a prescribed majorant-weight. Attention is called to a part of its proof that requires an additional argument. Such an argument, based on a new characterization of Beurling's lower uniform density, is then presented.