Trimmed sums for observables on the doubling map

We establish a strong law of large numbers under intermediate trimming for a particular example of Birkhoff sums of a non-integrable observable over the doubling map. It has been shown in a previous work by Haynes that there is no strong law of large numbers for the considered system after removing finitely many summands (light trimming) even though i.i.d. random variables and also some dynamical systems with the same distribution function obey a strong law of large numbers after removing only the largest summand.