  Chang's Lemma is a widely employed result in additive combinatorics. It gives
bounds on the dimension of the large spectrum of probability distributions on
finite abelian groups. Recently, Bloom (2016) presented a powerful variant of
Chang's Lemma that yields the strongest known quantitative version of Roth's
theorem on 3-term arithmetic progressions in dense subsets of the integers. In
this note, we show how such theorems can be derived from the approximation of
probability measures via entropy maximization.
