Equivalent norms in spaces of entire functions

It is shown that if $E\subset\mathbf R^n$ is relatively dense with respect to Lebesgue mesure and $p\in(0,\infty)$, then for any entire function $f(z)$ of $n$ complex variables and of exponential type not exceeding $\sigma$ the inequality
$$ \int_E|f(x)|^p\,dx_1\dots dx_n\geqslant c\int_{\mathbf R^n}|f(x)|^p\,dx_1\dots dx_n $$
is satisfied, where $c$ is a constant depending only on $\sigma$, $L$, $\delta$ and $p$, but not on $f(z)$, and the integrals on both sides of the inequality converge or diverge simultaneously.
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