Perhaps as simple as the white case reflect light better than the purple?
My black Z1 has also some kind of dim light I suppose... (i have not had any change to compare)
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Sony android devices i currently own:
SONY Xperia X10i, X10 mini, X10 mini Pro, Arc, Ray, S, T, V, J, Z1
>> ! Don“t expect firmware miracles from SONY, or you will get disappointed ! <<