Schnirelmann density

Schnirelmann density   (Redirected from Mann's theorem) Jump to navigation Jump to search In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.[1][2] Contents 1 Definition 2 Properties 2.1 Sensitivity 3 Schnirelmann's theorems 4 Additive bases 5 Mann's theorem 6 Waring's problem 7 Schnirelmann's constant 8 Essential components 9 References Definition The Schnirelmann density of a set of natural numbers A is defined as {displaystyle sigma A=inf _{n}{frac {A(n)}{n}},} where A(n) denotes the number of elements of A not exceeding n and inf is infimum.[3] The Schnirelmann density is well-defined even if the limit of A(n)/n as n → ∞ fails to exist (see upper and lower asymptotic 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.[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 {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.

Schnirelmann's constant In 1930 Schnirelmann used these ideas in conjunction with the Brun sieve to prove Schnirelmann's theorem,[1][2] that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant:[6] Schnirelmann obtained C < 800000.[7] Schnirelmann's constant is the lowest number C with this property.[6] Olivier Ramaré showed in (Ramaré 1995) that Schnirelmann's constant is at most 7,[6] improving the earlier upper bound of 19 obtained by Hans Riesel and R. C. Vaughan. Schnirelmann's constant is at least 3; Goldbach's conjecture implies that this is the constant's actual value.[6] In 2013, Harald Helfgott proved Goldbach's weak conjecture for all odd numbers. Therefore Schnirelmann's constant is at most 4. [8][9][10][11] Essential components Khintchin proved that the sequence of squares, though of zero Schnirelmann density, when added to a sequence of Schnirelmann density between 0 and 1, increases the density: {displaystyle sigma (A+{mathfrak {G}}^{2})>sigma (A){text{ for }}0 1, and for every c > 1 there is an essential component which has at most (log x)c elements up to x.[15] References ^ Jump up to: a b Schnirelmann, L.G. (1930). "On the additive properties of numbers", first published in "Proceedings of the Don Polytechnic Institute in Novocherkassk" (in Russian), vol XIV (1930), pp. 3-27, and reprinted in "Uspekhi Matematicheskikh Nauk" (in Russian), 1939, no. 6, 9–25. ^ Jump up to: a b Schnirelmann, L.G. (1933). First published as "Über additive Eigenschaften von Zahlen" in "Mathematische Annalen" (in German), vol 107 (1933), 649-690, and reprinted as "On the additive properties of numbers" in "Uspekhin. Matematicheskikh Nauk" (in Russian), 1940, no. 7, 7–46. ^ Nathanson (1996) pp.191–192 ^ Nathanson (1990) p.397 ^ E. Artin and P. Scherk (1943) On the sums of two sets of integers, Ann. of Math 44, page=138-142. ^ Jump up to: a b c d Nathanson (1996) p.208 ^ Gelfond & Linnik (1966) p.136 ^ Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT]. ^ Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT]. ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT]. ^ Helfgoot, Harald A. (2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT]. ^ Ruzsa (2009) p.177 ^ Ruzsa (2009) p.179 ^ Linnik, Yu. V. (1942). "On Erdõs's theorem on the addition of numerical sequences". Mat. Sb. 10: 67–78. Zbl 0063.03574. ^ Ruzsa (2009) p.184 Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringsches Problem)". Mathematische Annalen. 67 (3): 281–300. doi:10.1007/BF01450405. ISSN 0025-5831. MR 1511530. S2CID 179177986. Schnirelmann, L.G. (1930). "On additive properties of numbers". Ann. Inst. Polytechn. Novočerkassk (in Russian). 14: 3–28. JFM 56.0892.02. Schnirelmann, L.G. (1933). "Über additive Eigenschaften von Zahlen". Math. Ann. (in German). 107: 649–690. doi:10.1007/BF01448914. S2CID 123067485. Zbl 0006.10402. Mann, Henry B. (1942). "A proof of the fundamental theorem on the density of sums of sets of positive integers". Annals of Mathematics. Second Series. 43 (3): 523–527. doi:10.2307/1968807. ISSN 0003-486X. JSTOR 1968807. MR 0006748. Zbl 0061.07406. Gelfond, A.O.; Linnik, Yu. V. (1966). L.J. Mordell (ed.). Elementary Methods in Analytic Number Theory. George Allen & Unwin. Mann, Henry B. (1976). Addition Theorems: The Addition Theorems of Group Theory and Number Theory (Corrected reprint of 1965 Wiley ed.). Huntington, New York: Robert E. Krieger Publishing Company. ISBN 978-0-88275-418-5. MR 0424744. {{cite book}}: External link in |publisher= (help) Nathanson, Melvyn B. (1990). "Best possible results on the density of sumsets". In Berndt, Bruce C.; Diamond, Harold G.; Halberstam, Heini; et al. (eds.). Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA). Progress in Mathematics. Vol. 85. Boston: Birkhäuser. pp. 395–403. ISBN 978-0-8176-3481-0. Zbl 0722.11007. Ramaré, O. (1995). "On Šnirel'man's constant". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 22 (4): 645–706. Zbl 0851.11057. Retrieved 2011-03-28. Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 978-0-387-94656-6. Zbl 0859.11002. Nathanson, Melvyn B. (2000). Elementary Methods in Number Theory. Graduate Texts in Mathematics. Vol. 195. Springer-Verlag. pp. 359–367. ISBN 978-0-387-98912-9. Zbl 0953.11002. Khinchin, A. Ya. (1998). Three Pearls of Number Theory. Mineola, NY: Dover. ISBN 978-0-486-40026-6. Has a proof of Mann's theorem and the Schnirelmann-density proof of Waring's conjecture. Artin, Emil; Scherk, P. (1943). "On the sums of two set of integers". Ann. of Math. 44: 138–142. Cojocaru, Alina Carmen; Murty, M. Ram (2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. pp. 100–105. ISBN 978-0-521-61275-3. Ruzsa, Imre Z. (2009). "Sumsets and structure". In Geroldinger, Alfred; Ruzsa, Imre Z. (eds.). Combinatorial number theory and additive group theory. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. pp. 87–210. ISBN 978-3-7643-8961-1. Zbl 1221.11026. Categories: Additive number theoryMathematical constants

Si quieres conocer otros artículos parecidos a Schnirelmann density puedes visitar la categoría Additive number theory.

Deja una respuesta

Tu dirección de correo electrónico no será publicada.


Utilizamos cookies propias y de terceros para mejorar la experiencia de usuario Más información