Schnirelmann density

Properties By definition, 0 ≤ A(n) ≤ n and n σA ≤ A(n) for all n, und deshalb 0 ≤ σA ≤ 1, and σA = 1 if and only if A = N. Außerdem, {displaystyle sigma A=0Rightarrow forall epsilon >0 exists n A(n)0} then there exists {Anzeigestil k} so dass {Anzeigestil 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. Daher, the last theorem states that any set with positive Schnirelmann density is an additive basis. In dieser Terminologie, the set of squares {Anzeigestil {mathfrak {G}}^{2}={k^{2}}_{k=1}^{unendlich }} 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.

Satz. (Mann 1942) Lassen {Anzeigestil A} und {Anzeigestil B} be subsets of {Anzeigestil mathbb {N} } . In case that {displaystyle Aoplus Bneq mathbb {N} } , Wir haben noch {Display-Sigma (Aoplus B)geq sigma A+sigma B.} An analogue of this theorem for lower asymptotic density was obtained by Kneser.[4] At a later date, E. Artin and P. Scherk simplified the proof of Mann's theorem.[5] Waring's problem Main article: Waring's problem Let {Anzeigestil k} und {Anzeigestil N} be natural numbers. Lassen {Anzeigestil {mathfrak {G}}^{k}={i^{k}}_{i=1}^{unendlich }} . Definieren {Anzeigestil r_{N}^{k}(n)} to be the number of non-negative integral solutions to the equation {Anzeigestil x_{1}^{k}+x_{2}^{k}+cdots +x_{N}^{k}=n} und {Anzeigestil 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 {Anzeigestil x_{ich}} , beziehungsweise. Daher {Anzeigestil R_{N}^{k}(n)= Summe _{ich=0}^{n}r_{N}^{k}(ich)} . Wir haben {Anzeigestil r_{N}^{k}(n)>0leftrightarrow nin N{mathfrak {G}}^{k},} {Anzeigestil R_{N}^{k}(n)geklinkt({frac {n}{N}}Rechts)^{frac {N}{k}}.} The volume of the {Anzeigestil 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 {Anzeigestil n^{1/k}} , somit {Anzeigestil R_{N}^{k}(n)= Summe _{ich=0}^{n}r_{N}^{k}(ich)leq n^{N/k}} . The hard part is to show that this bound still works on the average, d.h., Lemma. (Linnik) Für alle {Anzeigestil kin mathbb {N} } es existiert {Anzeigestil Nin mathbb {N} } and a constant {displaystyle c=c(k)} , depending only on {Anzeigestil k} , so dass für alle {Anzeigestil nin mathbb {N} } , {Anzeigestil r_{N}^{k}(m)0} .

We have thus established the general solution to Waring's Problem: Logische Folge. (Hilbert 1909) Für alle {Anzeigestil k} es existiert {Anzeigestil N} , depending only on {Anzeigestil k} , such that every positive integer {Anzeigestil n} can be expressed as the sum of at most {Anzeigestil N} many {Anzeigestil k} -th powers.