# Schnirelmann density Properties By definition, 0 ≤ A(n) ≤ n and n σA ≤ A(n) for all n, and therefore 0 ≤ σA ≤ 1, and σA = 1 if and only if A = N. Furthermore, {displaystyle sigma A=0Rightarrow forall epsilon >0 exists n A(n)0} then there exists {displaystyle k} such that {displaystyle bigoplus _{i=1}^{k}A=mathbb {N} .} Additive bases A subset {displaystyle Asubseteq mathbb {N} } with the property that {displaystyle Aoplus Aoplus cdots oplus A=mathbb {N} } for a finite sum, is called an additive basis, and the least number of summands required is called the degree (sometimes order) of the basis. Thus, the last theorem states that any set with positive Schnirelmann density is an additive basis. In this terminology, the set of squares {displaystyle {mathfrak {G}}^{2}={k^{2}}_{k=1}^{infty }} is an additive basis of degree 4. (About an open problem for additive bases, see Erdős–Turán conjecture on additive bases.) Mann's theorem Historically the theorems above were pointers to the following result, at one time known as the {displaystyle alpha +beta } hypothesis. It was used by Edmund Landau and was finally proved by Henry Mann in 1942.

Theorem. (Mann 1942) Let {displaystyle A} and {displaystyle B} be subsets of {displaystyle mathbb {N} } . In case that {displaystyle Aoplus Bneq mathbb {N} } , we still have {displaystyle sigma (Aoplus B)geq sigma A+sigma B.} An analogue of this theorem for lower asymptotic density was obtained by Kneser. At a later date, E. Artin and P. Scherk simplified the proof of Mann's theorem. Waring's problem Main article: Waring's problem Let {displaystyle k} and {displaystyle N} be natural numbers. Let {displaystyle {mathfrak {G}}^{k}={i^{k}}_{i=1}^{infty }} . Define {displaystyle r_{N}^{k}(n)} to be the number of non-negative integral solutions to the equation {displaystyle x_{1}^{k}+x_{2}^{k}+cdots +x_{N}^{k}=n} and {displaystyle R_{N}^{k}(n)} to be the number of non-negative integral solutions to the inequality {displaystyle 0leq x_{1}^{k}+x_{2}^{k}+cdots +x_{N}^{k}leq n,} in the variables {displaystyle x_{i}} , respectively. Thus {displaystyle R_{N}^{k}(n)=sum _{i=0}^{n}r_{N}^{k}(i)} . We have {displaystyle r_{N}^{k}(n)>0leftrightarrow nin N{mathfrak {G}}^{k},} {displaystyle R_{N}^{k}(n)geq left({frac {n}{N}}right)^{frac {N}{k}}.} The volume of the {displaystyle N} -dimensional body defined by {displaystyle 0leq x_{1}^{k}+x_{2}^{k}+cdots +x_{N}^{k}leq n} , is bounded by the volume of the hypercube of size {displaystyle n^{1/k}} , hence {displaystyle R_{N}^{k}(n)=sum _{i=0}^{n}r_{N}^{k}(i)leq n^{N/k}} . The hard part is to show that this bound still works on the average, i.e., Lemma. (Linnik) For all {displaystyle kin mathbb {N} } there exists {displaystyle Nin mathbb {N} } and a constant {displaystyle c=c(k)} , depending only on {displaystyle k} , such that for all {displaystyle nin mathbb {N} } , {displaystyle r_{N}^{k}(m)0} .

We have thus established the general solution to Waring's Problem: Corollary. (Hilbert 1909) For all {displaystyle k} there exists {displaystyle N} , depending only on {displaystyle k} , such that every positive integer {displaystyle n} can be expressed as the sum of at most {displaystyle N} many {displaystyle k} -th powers.