Algebraic equations with lacunary polynomials and the Erdos-Renyi conjecture

In 1947, Rényi, Kalmár and Rédei discovered some special polynomials p(x)∈C[x] for which the square p(x)2 has fewer non-zero terms than p(x). Rényi and Erdős then conjectured that if the number of terms of p(x) grows to infinity, then the same happens for p(x)2. The conjecture was later proved by Schinzel, strengthened by Zannier, and a 'final' generalisation was proved by C. Fuchs, Zannier and the author. This note is a survey of the known results, with a focus on the applications of the latest generalisation.