Entropies of weighted sums in cyclic groups and an application to polar codes

Understand

In this note, the following basic question is explored: in a cyclic group, how are the Shannon entropies of the sum and difference of i.i.d.

• random variables related to each other? For the integer group, we show that they can differ by any real number additively, but not too much multiplicatively; on the other hand, for $\mathbb{Z}/3\mathbb{Z}$, the entropy of the difference is always at least as large as that of the sum.
• These results are closely related to the study of more-sum-than-difference (i.e.
• MSTD) sets in additive combinatorics.