Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure
Abstract We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that 
$|f(z)|\ge \exp (|z|^\alpha )$
 for some 
$\alpha>0$
 and all z outside a set of finite Lebesgue measure.