The entropies of the sum and the difference of two IID random variables are not too different

Consider the entropy increase h(Y + Y´) - h(Y) of the sum of two continuous i.i.d. random variables Y, Y´, and the corresponding entropy increase h(Y - Y´) - h(Y) of their difference. We show that the ratio between these two quantities always lies between 1/2 and 2. This complements a recent result of Lapidoth and Pete, showing that the difference h(Y + Y´) - h(Y - Y´) may be arbitrarily large. Corresponding results are discussed for the discrete entropy, and connections are drawn with exciting recent mathematical work in the area of additive combinatorics.