Additive Representation in Thin Sequences, IV: Lower Bound Methods

We describe a method for establishing that values from a fixed polynomial sequence are represented frequently by some prescribed sum of powers of natural numbers. As an illustration of this method, we show that for at least X129/136 of the integers n with 1 ≤ n ≤ X, a fixed quadratic polynomial φ(n) may be written as the sum of five cubes of positive integers. A similar result is established for the sum of a square and three cubes of positive integers.