Algebraic equations with lacunary polynomials and the Erdos-Renyi conjecture

In 1947, Renyi, Kalmar and Redei discovered some special polynomials p(x) is an element of C[x] for which the square p(x)(2) has fewer non-zero terms than p(x). Renyi and Erdos 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.