Skip to main content

2013 | OriginalPaper | Buchkapitel

On Additive Representative Functions

verfasst von : András Sárközy, Vera T. Sós

Erschienen in: The Mathematics of Paul Erdős I

Verlag: Springer New York

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper we give a short survey of additive representation functions, in particular, on their regularity properties and value distribution. We prove a couple of new results and present many related unsolved problems.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Literatur
1.
Zurück zum Zitat M. Ajtai, J. Komlós, E. Szemerédi: A dense infinite Sidon sequence, European J. Comb. 2 (1981), 1–11.MATH M. Ajtai, J. Komlós, E. Szemerédi: A dense infinite Sidon sequence, European J. Comb. 2 (1981), 1–11.MATH
2.
Zurück zum Zitat R. Balasubramanian, A note on a result of Erdős, Sárközy and Sós, Acta Arithmetica 49 (1987), 45–53.MATH R. Balasubramanian, A note on a result of Erdős, Sárközy and Sós, Acta Arithmetica 49 (1987), 45–53.MATH
3.
Zurück zum Zitat P. T. Bateman, E. E. Kohlbecker and J. P. Tull, On a theorem of Erdős and Fuchs in additive number theory, Proc. Amer. Math. Soc. 14 (1963), 278–284.MathSciNetMATH P. T. Bateman, E. E. Kohlbecker and J. P. Tull, On a theorem of Erdős and Fuchs in additive number theory, Proc. Amer. Math. Soc. 14 (1963), 278–284.MathSciNetMATH
4.
Zurück zum Zitat B. Bollobás, Random graphs, Academic, New York, 1985.MATH B. Bollobás, Random graphs, Academic, New York, 1985.MATH
5.
Zurück zum Zitat S. Chowla, Solution of a problem of Erdős and Turán in additive number theory, Proc. Nat. Acad. Sci. India 14 (1944), 1–2.MathSciNetMATH S. Chowla, Solution of a problem of Erdős and Turán in additive number theory, Proc. Nat. Acad. Sci. India 14 (1944), 1–2.MathSciNetMATH
7.
Zurück zum Zitat P. Erdős, Addendum, On a problem of Sidon in additive number theory and on some related problems, J. London Math. Soc. 19 (1944), 208.CrossRef P. Erdős, Addendum, On a problem of Sidon in additive number theory and on some related problems, J. London Math. Soc. 19 (1944), 208.CrossRef
8.
Zurück zum Zitat P. Erdős, Problems and results in additive number theory, Colloque sur la Théorie des Nombres (CBRM) (Bruxelles, 1956), 127–137. P. Erdős, Problems and results in additive number theory, Colloque sur la Théorie des Nombres (CBRM) (Bruxelles, 1956), 127–137.
10.
Zurück zum Zitat P. Erdős, On some applications of graph theory to number theory, Publ. Ramanujan Inst. 1 (1969), 131–136. P. Erdős, On some applications of graph theory to number theory, Publ. Ramanujan Inst. 1 (1969), 131–136.
11.
Zurück zum Zitat P. Erdős and R. Freud, On Sidon sequences and related problems, Mat. Lapok 1 (1991), 1–44 (in Hungarian). P. Erdős and R. Freud, On Sidon sequences and related problems, Mat. Lapok 1 (1991), 1–44 (in Hungarian).
12.
Zurück zum Zitat P. Erdős and W. H. J. Fuchs, On a problem of additive number theory, J. London Math. Soc. 31 (1956), 67–73.MathSciNetCrossRef P. Erdős and W. H. J. Fuchs, On a problem of additive number theory, J. London Math. Soc. 31 (1956), 67–73.MathSciNetCrossRef
13.
Zurück zum Zitat P. Erdős and A. Rényi, Additive properties of random sequences of positive integers, Acta Arithmetica 6 (1960), 83–110.MathSciNet P. Erdős and A. Rényi, Additive properties of random sequences of positive integers, Acta Arithmetica 6 (1960), 83–110.MathSciNet
14.
Zurück zum Zitat P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, I, Pacific J. 118 (1985), 347–357.CrossRef P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, I, Pacific J. 118 (1985), 347–357.CrossRef
15.
Zurück zum Zitat P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, II, Acta Math. Hung. 48 (1986), 201–211.CrossRef P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, II, Acta Math. Hung. 48 (1986), 201–211.CrossRef
16.
Zurück zum Zitat P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, III, Studia Sci. Math. Hung. 22 (1987), 53–63. P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, III, Studia Sci. Math. Hung. 22 (1987), 53–63.
17.
Zurück zum Zitat P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, IV, in: Number Theory, Proceedings, Ootacamund, India, 1984, Lecture Notes in Mathematics 1122, Springer-Verlag, 1985; 85–104. P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, IV, in: Number Theory, Proceedings, Ootacamund, India, 1984, Lecture Notes in Mathematics 1122, Springer-Verlag, 1985; 85–104.
18.
Zurück zum Zitat P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, V, Monatshefte Math. 102 (1986), 183–197.CrossRef P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, V, Monatshefte Math. 102 (1986), 183–197.CrossRef
19.
20.
Zurück zum Zitat P. Erdős, A. Sárközy and V. T. Sós, On sum sets of Sidon sets, II, Israel J. Math. 90 (1995), 221–233.MathSciNetCrossRef P. Erdős, A. Sárközy and V. T. Sós, On sum sets of Sidon sets, II, Israel J. Math. 90 (1995), 221–233.MathSciNetCrossRef
21.
Zurück zum Zitat P. Erdős, A. Sárközy and V. T. Sós, On additive properties of general sequences, Discrete Math. 136 (1994), 75–99.MathSciNetCrossRef P. Erdős, A. Sárközy and V. T. Sós, On additive properties of general sequences, Discrete Math. 136 (1994), 75–99.MathSciNetCrossRef
22.
Zurück zum Zitat P. Erdős and P. Turán, On a problem of Sidon in additive number theory and some related problems, J. London Math. Soc. 16 (1941), 212–215.MathSciNetCrossRef P. Erdős and P. Turán, On a problem of Sidon in additive number theory and some related problems, J. London Math. Soc. 16 (1941), 212–215.MathSciNetCrossRef
23.
24.
Zurück zum Zitat E. K. Hayashi, An elementary method for estimating error terms in additive number theory, Proc. Amer. Math. Soc. 52 (1975), 55–59.MathSciNetMATHCrossRef E. K. Hayashi, An elementary method for estimating error terms in additive number theory, Proc. Amer. Math. Soc. 52 (1975), 55–59.MathSciNetMATHCrossRef
25.
Zurück zum Zitat E. K. Hayashi, Omega theorems for the iterated additive convolution of a nonnegative arithmetic function, J. Number Theory 13 (1981), 176–191.MathSciNetMATHCrossRef E. K. Hayashi, Omega theorems for the iterated additive convolution of a nonnegative arithmetic function, J. Number Theory 13 (1981), 176–191.MathSciNetMATHCrossRef
27.
Zurück zum Zitat A. Gyárfás, Z. Lehel, Linear sets with five distinct differences among any four elements, J. Combin. Theory Ser. B 64 (1995), 108–118.MathSciNetMATHCrossRef A. Gyárfás, Z. Lehel, Linear sets with five distinct differences among any four elements, J. Combin. Theory Ser. B 64 (1995), 108–118.MathSciNetMATHCrossRef
28.
Zurück zum Zitat H. L. Montgomery and R. C. Vaughan, On the Erdős-Fuchs theorems, in: A tribute to Paul Erdős, eds. A. Baker, B. Bollobás and A. Hajnal, Cambridge Univ. Press, 1990; 331–338. H. L. Montgomery and R. C. Vaughan, On the Erdős-Fuchs theorems, in: A tribute to Paul Erdős, eds. A. Baker, B. Bollobás and A. Hajnal, Cambridge Univ. Press, 1990; 331–338.
29.
34.
35.
Zurück zum Zitat J. Schur, Über die Kongruenz \({x}^{m} + {y}^{m} \equiv {z}^{m}\ ({\rm mod}\ p)\), Jahresbericht der Deutschen Math. Verein. 25 (1916), 114–117.MATH J. Schur, Über die Kongruenz \({x}^{m} + {y}^{m} \equiv {z}^{m}\ ({\rm mod}\ p)\), Jahresbericht der Deutschen Math. Verein. 25 (1916), 114–117.MATH
36.
Zurück zum Zitat S. Sidon, Ein Satz über trigonomische Polynome und seine Anwendung in der Theorie der Fourier-Reihen, Math. Annalen 106 (1932), 536–539.MathSciNetCrossRef S. Sidon, Ein Satz über trigonomische Polynome und seine Anwendung in der Theorie der Fourier-Reihen, Math. Annalen 106 (1932), 536–539.MathSciNetCrossRef
37.
Zurück zum Zitat V.T. Sós, An additive problem on different structures, 3rd Internat. Comb. Conf., San Francisco 1989. Graph Theory, Comb. Alg. and Appl. SIAM, ed. Y. Alavi, F. R. K. Chung, R. L. Graham, D. F. Hsu (1991), 486–508. V.T. Sós, An additive problem on different structures, 3rd Internat. Comb. Conf., San Francisco 1989. Graph Theory, Comb. Alg. and Appl. SIAM, ed. Y. Alavi, F. R. K. Chung, R. L. Graham, D. F. Hsu (1991), 486–508.
38.
Zurück zum Zitat A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlen, J. Reine Angew. Math. 194 (1955), 40–65, 111–140.MathSciNetMATH A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlen, J. Reine Angew. Math. 194 (1955), 40–65, 111–140.MathSciNetMATH
39.
Zurück zum Zitat E. Szemerédi, On a set containing no k elements in an arithmetic progression, Acta Arithmetica 27 (1975), 199–245.MathSciNetMATH E. Szemerédi, On a set containing no k elements in an arithmetic progression, Acta Arithmetica 27 (1975), 199–245.MathSciNetMATH
41.
Zurück zum Zitat B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212–216.MATH B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212–216.MATH
42.
Zurück zum Zitat L. Babai and V. T. Sós, Sidon sets in groups and induced subgraphs of Cayley graphs, European J. Combin. 6 (1985), 101–114.MathSciNetMATH L. Babai and V. T. Sós, Sidon sets in groups and induced subgraphs of Cayley graphs, European J. Combin. 6 (1985), 101–114.MathSciNetMATH
43.
Zurück zum Zitat J. Borbély, On the higher dimensional generalization of a problem of Roth, Integers, to appear. J. Borbély, On the higher dimensional generalization of a problem of Roth, Integers, to appear.
44.
46.
47.
48.
Zurück zum Zitat Y.-G. Chen and M. Tang, On the monotonicity properties of additive representation functions, II, Discrete Math. 309 (2009), 1368–1373.MathSciNetMATHCrossRef Y.-G. Chen and M. Tang, On the monotonicity properties of additive representation functions, II, Discrete Math. 309 (2009), 1368–1373.MathSciNetMATHCrossRef
49.
Zurück zum Zitat Y.-G. Chen and M. Tang, Some extension of a property of linear representation functions, Discrete Math. 309 (2009), 6294–6298.MathSciNetMATHCrossRef Y.-G. Chen and M. Tang, Some extension of a property of linear representation functions, Discrete Math. 309 (2009), 6294–6298.MathSciNetMATHCrossRef
50.
Zurück zum Zitat Y.-G. Chen, A. Sárközy, V. T. Sós and M. Tang, On the monotonicity properties of additive representation functions, Bull. Austral. Math. Soc. 72 (2005), 129–138.MathSciNetMATHCrossRef Y.-G. Chen, A. Sárközy, V. T. Sós and M. Tang, On the monotonicity properties of additive representation functions, Bull. Austral. Math. Soc. 72 (2005), 129–138.MathSciNetMATHCrossRef
52.
Zurück zum Zitat J. Cilleruelo, S. Kiss, I. Z. Ruzsa and C. Vinuesa, Generalization of a theorem of Erdős and Rényi on Sidon sequences, Random Structures Algorithms 37 (2010), 455–464.MathSciNetMATHCrossRef J. Cilleruelo, S. Kiss, I. Z. Ruzsa and C. Vinuesa, Generalization of a theorem of Erdős and Rényi on Sidon sequences, Random Structures Algorithms 37 (2010), 455–464.MathSciNetMATHCrossRef
53.
Zurück zum Zitat P. Csikvári, K. Gyarmati and A. Sárközy, Density and Ramsey-type results on algebraic equations with restricted solution sets, Combinatorica 32 (2012), 425–449.MathSciNetCrossRef P. Csikvári, K. Gyarmati and A. Sárközy, Density and Ramsey-type results on algebraic equations with restricted solution sets, Combinatorica 32 (2012), 425–449.MathSciNetCrossRef
55.
Zurück zum Zitat A. Dubickas, A basis of finite and infinite sets with small representation function, Electron. J. Combin. 19 (2012), Paper 6, 16 pp. A. Dubickas, A basis of finite and infinite sets with small representation function, Electron. J. Combin. 19 (2012), Paper 6, 16 pp.
56.
Zurück zum Zitat P. Erdős and A. Sárközy, On a conjecture of Roth and some related problems, II, in: Number Theory, Proc. of the First Conference of the Canadian Number Theory Association, ed. R. A. Mollin, Walter de Gruyter, Berlin–New York, 1990; 125–138. P. Erdős and A. Sárközy, On a conjecture of Roth and some related problems, II, in: Number Theory, Proc. of the First Conference of the Canadian Number Theory Association, ed. R. A. Mollin, Walter de Gruyter, Berlin–New York, 1990; 125–138.
57.
Zurück zum Zitat P. Erdős and P. Tetali, Representations of integers as the sum of k terms, Random Struct. Algorithms 1 (1990), 245–261.CrossRef P. Erdős and P. Tetali, Representations of integers as the sum of k terms, Random Struct. Algorithms 1 (1990), 245–261.CrossRef
58.
Zurück zum Zitat P. Erdős, M. B. Nathanson and P. Tetali, Independence of solution sets and minimal asymptotic bases, Acta Arith. 69 (1995), 243–258.MathSciNet P. Erdős, M. B. Nathanson and P. Tetali, Independence of solution sets and minimal asymptotic bases, Acta Arith. 69 (1995), 243–258.MathSciNet
59.
Zurück zum Zitat P. Erdős, A. Sárközy and V. T. Sós, On a conjecture of Roth and some related problems, I, in: Irregularities of Partitions, eds. G. Halász and V. T. Sós, Algorithms and Combinatorics 8, Springer, 1989; 47–59. P. Erdős, A. Sárközy and V. T. Sós, On a conjecture of Roth and some related problems, I, in: Irregularities of Partitions, eds. G. Halász and V. T. Sós, Algorithms and Combinatorics 8, Springer, 1989; 47–59.
60.
61.
Zurück zum Zitat K. Gyarmati, F. Hennecart and I. Z. Ruzsa, Sums and differences of finite sets, Funct. Approx. Comment. Math. 37 (2007), 157–186.MathSciNetCrossRef K. Gyarmati, F. Hennecart and I. Z. Ruzsa, Sums and differences of finite sets, Funct. Approx. Comment. Math. 37 (2007), 157–186.MathSciNetCrossRef
62.
Zurück zum Zitat L. Haddad and C. Helou, Additive bases representations in groups, Integers 8 (2008), A5, 9 pp. L. Haddad and C. Helou, Additive bases representations in groups, Integers 8 (2008), A5, 9 pp.
63.
Zurück zum Zitat N. Hindman, Monochromatic sums equal to products in \(\mathbb{N}\), Integers 11A (2011), Art. 10, 1–10. N. Hindman, Monochromatic sums equal to products in \(\mathbb{N}\), Integers 11A (2011), Art. 10, 1–10.
64.
66.
67.
Zurück zum Zitat G. Horváth, On a property of linear representation functions, Studia Sci. Math. Hungar. 39 (2002), 203–214.MathSciNetMATH G. Horváth, On a property of linear representation functions, Studia Sci. Math. Hungar. 39 (2002), 203–214.MathSciNetMATH
69.
71.
72.
Zurück zum Zitat J. Konstantoulas, Laver bounds for a conjecture of Erdős and Turán, Acta Arith., to appear. J. Konstantoulas, Laver bounds for a conjecture of Erdős and Turán, Acta Arith., to appear.
73.
Zurück zum Zitat S. Konyagin and V. T. Lev, The Erdős-Turán problem in infinite groups, in: Additive number theory, 195–202, Springer, New York, 2010. S. Konyagin and V. T. Lev, The Erdős-Turán problem in infinite groups, in: Additive number theory, 195–202, Springer, New York, 2010.
74.
Zurück zum Zitat V. F. Lev and A. Sárközy, An Erdős–Fuchs-type theorem for finite groups, Integers 11A (2011), Art. 15, 7 p. V. F. Lev and A. Sárközy, An Erdős–Fuchs-type theorem for finite groups, Integers 11A (2011), Art. 15, 7 p.
75.
Zurück zum Zitat G. Martin and K. O’Bryant, Many sets have more sums than differences, in: CRM Proceedings and Lecture Notes, vol. 43, 2007; 287–305. G. Martin and K. O’Bryant, Many sets have more sums than differences, in: CRM Proceedings and Lecture Notes, vol. 43, 2007; 287–305.
76.
Zurück zum Zitat M. B. Nathanson, Sets with more sums than differences, Integers 7 (2007), A5, 24 pp. M. B. Nathanson, Sets with more sums than differences, Integers 7 (2007), A5, 24 pp.
77.
Zurück zum Zitat K. O’Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin. Dynamic Survey 11 (2004), 39. K. O’Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin. Dynamic Survey 11 (2004), 39.
78.
Zurück zum Zitat P. P. Pach, Ramsey-type results on the solvability of certain equations in \(\mathbb{Z}_{m}\), Integers, to appear. P. P. Pach, Ramsey-type results on the solvability of certain equations in \(\mathbb{Z}_{m}\), Integers, to appear.
79.
Zurück zum Zitat P. P. Pach, The Ramsey-type version of a problem of Pomerance and Schinzel, Acta Arith., to appear. P. P. Pach, The Ramsey-type version of a problem of Pomerance and Schinzel, Acta Arith., to appear.
80.
Zurück zum Zitat C. Pomerance and A. Schinzel, Multiplicative properties of sets of residues, Moscow Math.J. Combin. Number Theory 1 (2011), 52–66.MathSciNetMATH C. Pomerance and A. Schinzel, Multiplicative properties of sets of residues, Moscow Math.J. Combin. Number Theory 1 (2011), 52–66.MathSciNetMATH
85.
Zurück zum Zitat I. Z. Ruzsa, Many differences, few sums, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 51 (2008), 27–38.MathSciNetMATH I. Z. Ruzsa, Many differences, few sums, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 51 (2008), 27–38.MathSciNetMATH
86.
Zurück zum Zitat C. Sándor, A note on a conjecture of Erdős-Turán, Integers 8 (2008), A30, 4 pp. C. Sándor, A note on a conjecture of Erdős-Turán, Integers 8 (2008), A30, 4 pp.
88.
Zurück zum Zitat A. Sárközy, A localized Erdős–Fuchs theorem, in: Bonner Mathematische Schriften, Nr. 360, Proceedings of the Session in analytic number theory and Diophantine equations (Bonn, January–June 2002), eds. D. R. Heath-Brown and B. Z. Moroz, Bonn, 2003. A. Sárközy, A localized Erdős–Fuchs theorem, in: Bonner Mathematische Schriften, Nr. 360, Proceedings of the Session in analytic number theory and Diophantine equations (Bonn, January–June 2002), eds. D. R. Heath-Brown and B. Z. Moroz, Bonn, 2003.
89.
Zurück zum Zitat A. Sárközy, On additive representation functions of finite sets, I (Variation), Period. Math. Hungar., to appear. A. Sárközy, On additive representation functions of finite sets, I (Variation), Period. Math. Hungar., to appear.
90.
Zurück zum Zitat I. D. Shkredev, On monochromatic solutions of some nonlinear equations in \(\mathbb{Z}/p\mathbb{Z}\) (Russian), Mat. Zametki 88 (2010), 603–611. I. D. Shkredev, On monochromatic solutions of some nonlinear equations in \(\mathbb{Z}/p\mathbb{Z}\) (Russian), Mat. Zametki 88 (2010), 603–611.
91.
Zurück zum Zitat J. Spencer and P. Tetali, Sidon sets with small gaps, in: Discrete probability and algorithms (Minneapolis, MN, 1993), 103–109, IMA Vol. Math. Appl. 72, Springer, New York, 1995. J. Spencer and P. Tetali, Sidon sets with small gaps, in: Discrete probability and algorithms (Minneapolis, MN, 1993), 103–109, IMA Vol. Math. Appl. 72, Springer, New York, 1995.
92.
Zurück zum Zitat M. Tang, A note on a result of Ruzsa, II, Bull. Austral. Math. Soc. 82 (2010), 340–347.MATHCrossRef M. Tang, A note on a result of Ruzsa, II, Bull. Austral. Math. Soc. 82 (2010), 340–347.MATHCrossRef
94.
Metadaten
Titel
On Additive Representative Functions
verfasst von
András Sárközy
Vera T. Sós
Copyright-Jahr
2013
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-7258-2_16