Davenport's problem on set addition

Let k be a natural number and A be a subset of the natural numbers. Can one determine the lower exponential density of the set
$$\{ x^k + a:  a\in A, x\in\mathbb N \}$$
in terms of lower exponential density of A? This problem is trivial for k=1, and has an elementary complete solution when k=2. Eighty years ago Davenport gave lower bounds in all cases of interest. In this talk we describe joint work with Simon Myerson. Davenport's estimates are improved when A is already fairly dense. In some range, our results are best possible.