In the paper On The Closure of Characters and the Zeros of Entire Functions by Beurling and Malliavin they make the following claim in the introduction.
The closure radius $\rho = \rho(\Lambda)$ defined as the upper bound of the numbers $r$ such that set $\{e^{i \lambda x}\}_{\lambda \in \Lambda}$ span the space $L^2(-r, r)$ (by span we mean that the span of the set $\{e^{i\lambda t}\}_{\lambda \in \Lambda} \text{is dense in} L^2(-r, r))$. The claim is that $\rho(\Lambda)$ does not change if the metric is replaced with any other $L^p$ metric.
In other words, if I understand correctly the claim is that $\rho(\Lambda)$ is independent of $p$ in $L^p$. How can be true? Surely, the topology should affect this somehow.