Expander estimates for cubes

Journal article, 2026

If A is a set of natural numbers of exponential density 8, then the exponential density of all numbers of the form x(3) + a with x is an element of N and alpha is an element of A is at least min(1, (1)/3 + 5/(6)delta). This is a considerable improvement on the previous best lower bounds for this problem, obtained by Davenport more than 80 years ago. The result is the best possible for delta > (4) /5 .