Schnirelmann density and Waring's problem

A central question in additive number theory is determining whether a given infinite set $A$ of positive integers is an asymptotic basis of finite order, meaning that every large enough integer is the sum of at most $C$ elements of $A$ for some constant $C$. Questions of this type of course cover the Goldbach conjectures, Waring's problem on representing numbers as sums of perfect $k$th powers, and Romanoff's theorem on writing an integer as the sum of $C$ primes and $C$ powers of two. A natural question is whether there is some relatively easily verifiable property of sets such that when this property holds, a set is an asymptotic basis. A natural candidate would be the positivity of the asymptotic density of the set, denoted by $d(A)$. This criterion could be used, but the asymptotic density fails to exist for some rather elementary sets, and even when it does exist, proving the existence can be hard for non-trivial sets. In 1930, L.G.Schnirelmann was able to define the right kind of density, called the Schnirelmann density and denoted by $\sigma(A)$, such that it always exists and any set with positive Schnirelmann density is an asymptotic basis of finite order. He went on to prove that every integer is the sum of $C$ primes for some $C$, and later Linnik used Schnirelmann's method to give a relatively simple solution to Waring's problem. It is also notable that Schnirelmann's and Linnik's proofs are quite elementary; the former one uses nothing more complicated than Brun's sieve (see this post), and the latter uses nothing more complicated than basic estimates for Weyl sums (see this post). We will prove in this post Schnirelmann's and Linnik's theorems, as well as Romanoff's theorem. We follow the book An Introduction to Sieve Methods and their Applications by Cojocaru and Murty and the book Not Always Buried Deep: A Second Course in Elementary Number Theory by Pollack.