Skip to main content
Top

2024 | OriginalPaper | Chapter

A Walk Through Some Newer Parts of Additive Combinatorics

Author : Béla Bajnok

Published in: Combinatorics, Graph Theory and Computing

Publisher: Springer Nature Switzerland

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this survey chapter we discuss some recent results and related open questions in additive combinatorics, in particular, questions about sumsets in finite abelian groups.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

Literature
1.
go back to reference N. Alon, M. B. Nathanson, and I. Ruzsa, Adding distinct congruence classes modulo a prime. Amer. Math. Monthly102 (1995), no. 3, 250–255.MathSciNetCrossRef N. Alon, M. B. Nathanson, and I. Ruzsa, Adding distinct congruence classes modulo a prime. Amer. Math. Monthly102 (1995), no. 3, 250–255.MathSciNetCrossRef
2.
go back to reference N. Alon, M. B. Nathanson, and I. Ruzsa, The polynomial method and restricted sums of congruence classes. J. Number Theory56 (1996), no. 2, 404–417.MathSciNetCrossRef N. Alon, M. B. Nathanson, and I. Ruzsa, The polynomial method and restricted sums of congruence classes. J. Number Theory56 (1996), no. 2, 404–417.MathSciNetCrossRef
3.
go back to reference B. Bajnok, On the maximum size of a \((k,l)\)-sumfree subset of an abelian group. Int. J. Number Theory5 (2009), no. 6, 953–971. B. Bajnok, On the maximum size of a \((k,l)\)-sumfree subset of an abelian group. Int. J. Number Theory5 (2009), no. 6, 953–971.
4.
go back to reference B. Bajnok, On the minimum size of restricted sumsets in cyclic groups. Acta Math. Hungar.148 (2016), no. 1, 228–256.MathSciNetCrossRef B. Bajnok, On the minimum size of restricted sumsets in cyclic groups. Acta Math. Hungar.148 (2016), no. 1, 228–256.MathSciNetCrossRef
5.
go back to reference B. Bajnok, The h-critical number of finite abelian groups. Unif. Distrib. Theory10 (2015), no.2, 93–15.MathSciNet B. Bajnok, The h-critical number of finite abelian groups. Unif. Distrib. Theory10 (2015), no.2, 93–15.MathSciNet
6.
go back to reference B. Bajnok, More on the h-critical numbers of finite abelian groups. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.59 (2016), 113–122.MathSciNet B. Bajnok, More on the h-critical numbers of finite abelian groups. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.59 (2016), 113–122.MathSciNet
7.
go back to reference B. Bajnok, Corrigendum to “The h-critical number of finite abelian groups.” Unif. Distrib. Theory12 (2017), no. 2, 119–124.MathSciNetCrossRef B. Bajnok, Corrigendum to “The h-critical number of finite abelian groups.” Unif. Distrib. Theory12 (2017), no. 2, 119–124.MathSciNetCrossRef
8.
go back to reference B. Bajnok, Open problems about sumsets in finite abelian groups: minimum sizes and critical numbers. Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer, New York, 2017, 9–23. B. Bajnok, Open problems about sumsets in finite abelian groups: minimum sizes and critical numbers. Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer, New York, 2017, 9–23.
9.
go back to reference B. Bajnok, Additive Combinatorics: A Menu of Research Problems. CRC Press, Boca Raton, 2018, xix+390 pp. B. Bajnok, Additive Combinatorics: A Menu of Research Problems. CRC Press, Boca Raton, 2018, xix+390 pp.
10.
go back to reference B. Bajnok, C. Berson, and H. A. Just, On Perfect Bases in Finite Abelian Groups. Involve15 (2022), No. 3, 525–536.MathSciNetCrossRef B. Bajnok, C. Berson, and H. A. Just, On Perfect Bases in Finite Abelian Groups. Involve15 (2022), No. 3, 525–536.MathSciNetCrossRef
11.
go back to reference B. Bajnok and R. Matzke, The minimum size of signed sumsets. Electron. J. Combin.22 (2015), no. 2, Paper 2.50, 17 pp. B. Bajnok and R. Matzke, The minimum size of signed sumsets. Electron. J. Combin.22 (2015), no. 2, Paper 2.50, 17 pp.
12.
go back to reference B. Bajnok and R. Matzke, On the minimum size of signed sumsets in elementary abelian groups. J. Number Theory159 (2016), 384–401.MathSciNetCrossRef B. Bajnok and R. Matzke, On the minimum size of signed sumsets in elementary abelian groups. J. Number Theory159 (2016), 384–401.MathSciNetCrossRef
13.
go back to reference B. Bajnok and R. Matzke, On the maximum size of \((k,l)\)-sumfree sets in cyclic groups. Bulletin of the Australian Mathematical Society99 (2019), no. 2, 184–194. B. Bajnok and R. Matzke, On the maximum size of \((k,l)\)-sumfree sets in cyclic groups. Bulletin of the Australian Mathematical Society99 (2019), no. 2, 184–194.
14.
go back to reference B. Bajnok and P. P. Pach, On sumsets of nonbases of maximum size. To appear in European Journal of Combinatorics. B. Bajnok and P. P. Pach, On sumsets of nonbases of maximum size. To appear in European Journal of Combinatorics.
15.
go back to reference B. Bajnok and I. Ruzsa, The independence number of a subset of an abelian group. Integers3 (2003), A2, 23 pp.MathSciNet B. Bajnok and I. Ruzsa, The independence number of a subset of an abelian group. Integers3 (2003), A2, 23 pp.MathSciNet
16.
go back to reference R. Balasubramanian, G. Prakash, and D. S. Ramana, Sum-free subsets of finite abelian groups of type III. European J. Combin.58 (2016), 181–202.MathSciNetCrossRef R. Balasubramanian, G. Prakash, and D. S. Ramana, Sum-free subsets of finite abelian groups of type III. European J. Combin.58 (2016), 181–202.MathSciNetCrossRef
17.
go back to reference B. C. Berndt, Y-S. Choi, and S-I. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society. Contemp. Math.236 (1999), 15–56.MathSciNetCrossRef B. C. Berndt, Y-S. Choi, and S-I. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society. Contemp. Math.236 (1999), 15–56.MathSciNetCrossRef
18.
go back to reference T. Bier and A. Y. M. Chin, On \((k,l)\)-sets in cyclic groups of odd prime order. Bull. Austral. Math. Soc.63 (2001), no. 1, 115–121. T. Bier and A. Y. M. Chin, On \((k,l)\)-sets in cyclic groups of odd prime order. Bull. Austral. Math. Soc.63 (2001), no. 1, 115–121.
19.
go back to reference A–L. Cauchy, Recherches sur les nombres. J. École Polytechnique9 (1813), 99–123. A–L. Cauchy, Recherches sur les nombres. J. École Polytechnique9 (1813), 99–123.
22.
go back to reference P. H. Diananda and H. P. Yap, Maximal sumfree sets of elements of finite groups. Proc. Japan Acad.45 (1969), 1–5.MathSciNet P. H. Diananda and H. P. Yap, Maximal sumfree sets of elements of finite groups. Proc. Japan Acad.45 (1969), 1–5.MathSciNet
23.
go back to reference J. A. Dias Da Silva and Y. O. Hamidoune, Cyclic space for Grassmann derivatives and additive theory. Bull. London Math. Soc.26 (1994), no. 2, 140–146.MathSciNetCrossRef J. A. Dias Da Silva and Y. O. Hamidoune, Cyclic space for Grassmann derivatives and additive theory. Bull. London Math. Soc.26 (1994), no. 2, 140–146.MathSciNetCrossRef
24.
go back to reference S. Eliahou and M. Kervaire, Sumsets in vector spaces over finite fields. J. Number Theory71 (1998), no. 1, 12–39.MathSciNetCrossRef S. Eliahou and M. Kervaire, Sumsets in vector spaces over finite fields. J. Number Theory71 (1998), no. 1, 12–39.MathSciNetCrossRef
25.
go back to reference P. Erdős, Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII pp. 181–189 Amer. Math. Soc., Providence, R.I., 1965. P. Erdős, Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII pp. 181–189 Amer. Math. Soc., Providence, R.I., 1965.
26.
go back to reference P. Erdős and R. L. Graham, On bases with an exact order. Acta Arith.37 (1980), 201–207. P. Erdős and R. L. Graham, On bases with an exact order. Acta Arith.37 (1980), 201–207.
27.
go back to reference P. Erdős and H. Heilbronn, On the addition of residue classes mod p. Acta Arith.9 (1964), 149–159. P. Erdős and H. Heilbronn, On the addition of residue classes mod p. Acta Arith.9 (1964), 149–159.
28.
go back to reference P. Erdős and M. B. Nathanson, Problems and results on minimal bases in additive number theory. Lecture Notes in Math.1240, Springer, Berlin (1987), 87–96. P. Erdős and M. B. Nathanson, Problems and results on minimal bases in additive number theory. Lecture Notes in Math.1240, Springer, Berlin (1987), 87–96.
29.
go back to reference P. Erdős and P. Turán, On a problem of Sidon in additive number theory and some related questions. J. London Math. Soc.16 (1941), 212–215. P. Erdős and P. Turán, On a problem of Sidon in additive number theory and some related questions. J. London Math. Soc.16 (1941), 212–215.
30.
go back to reference L. Gallardo, G. Grekos, L. Habsieger, F. Hennecart, B. Landreau, and A. Plagne, Restricted addition in \(\mathbb {Z}/n\mathbb {Z}\) and an application to the Erdős–Ginzburg–Ziv problem. J. London Math. Soc. (2)65 (2002), no. 3, 513–523. L. Gallardo, G. Grekos, L. Habsieger, F. Hennecart, B. Landreau, and A. Plagne, Restricted addition in \(\mathbb {Z}/n\mathbb {Z}\) and an application to the Erdős–Ginzburg–Ziv problem. J. London Math. Soc. (2)65 (2002), no. 3, 513–523.
31.
go back to reference B. Green and I. Ruzsa, sumfree sets in abelian groups. Israel J. Math.147 (2005), 157–188. B. Green and I. Ruzsa, sumfree sets in abelian groups. Israel J. Math.147 (2005), 157–188.
32.
go back to reference Y. O. Hamidoune and A. Plagne, A new critical pair theorem applied to sumfree sets in abelian groups. Comment. Math. Helv.79 (2004), no. 1, 183–207.MathSciNetCrossRef Y. O. Hamidoune and A. Plagne, A new critical pair theorem applied to sumfree sets in abelian groups. Comment. Math. Helv.79 (2004), no. 1, 183–207.MathSciNetCrossRef
33.
go back to reference Gy. Károlyi, On restricted set addition in abelian groups. Ann. Univ. Sci. Budapest, Eötvös Sect. Math.46, (2003) 47–54 (2004). Gy. Károlyi, On restricted set addition in abelian groups. Ann. Univ. Sci. Budapest, Eötvös Sect. Math.46, (2003) 47–54 (2004).
34.
go back to reference Gy. Károlyi, The Erdős–Heilbronn problem in abelian groups. Israel J. Math.139 (2004) 349–359. Gy. Károlyi, The Erdős–Heilbronn problem in abelian groups. Israel J. Math.139 (2004) 349–359.
35.
go back to reference Gy. Károlyi, An inverse theorem for the restricted set addition in abelian groups. J. Algebra290 (2005), no. 2, 557–593. Gy. Károlyi, An inverse theorem for the restricted set addition in abelian groups. J. Algebra290 (2005), no. 2, 557–593.
38.
39.
go back to reference V. F. Lev, Restricted set addition in groups. I. The classical setting. J. London Math. Soc. (2)62 (2000), no. 1, 27–40. V. F. Lev, Restricted set addition in groups. I. The classical setting. J. London Math. Soc. (2)62 (2000), no. 1, 27–40.
40.
go back to reference T. Nagell, Løsning til oppgave nr. 2, 1943, s. 29. Nordisk Mat. Tidskr.30 (1948), 62–64. T. Nagell, Løsning til oppgave nr. 2, 1943, s. 29. Nordisk Mat. Tidskr.30 (1948), 62–64.
41.
go back to reference T. Nagell, The Diophantine Equation \(x^2+7=2^n\). Ark. Mat.4 (1961), 185–187. T. Nagell, The Diophantine Equation \(x^2+7=2^n\). Ark. Mat.4 (1961), 185–187.
42.
go back to reference M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory. J. Number Theory6 (1974), 324–333.MathSciNetCrossRef M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory. J. Number Theory6 (1974), 324–333.MathSciNetCrossRef
43.
go back to reference M. B. Nathanson, Paul Erdős and additive bases. arXiv:1401.7598 [math.NT] (2014). M. B. Nathanson, Paul Erdős and additive bases. arXiv:1401.7598 [math.NT] (2014).
44.
go back to reference M. B. Nathanson, Additive number theory. Inverse problems and the geometry of sumsets. Graduate Texts in Mathematics, 165. Springer–Verlag, New York, 1996. xiv+293 pp. M. B. Nathanson, Additive number theory. Inverse problems and the geometry of sumsets. Graduate Texts in Mathematics, 165. Springer–Verlag, New York, 1996. xiv+293 pp.
45.
go back to reference A. Plagne, Maximal \((k,l)\)-free sets in \(\mathbb {Z}/p\mathbb {Z}\) are arithmetic progressions. Bull. Austral. Math. Soc.65 (2002), no. 3, 137–144. A. Plagne, Maximal \((k,l)\)-free sets in \(\mathbb {Z}/p\mathbb {Z}\) are arithmetic progressions. Bull. Austral. Math. Soc.65 (2002), no. 3, 137–144.
46.
go back to reference A. Plagne, Optimally small sumsets in groups, I. The supersmall sumset property, the \(\mu _G^{(k)}\) and the \(\nu _G^{(k)}\) functions. Unif. Distrib. Theory1 (2006), no. 1, 27–44. A. Plagne, Optimally small sumsets in groups, I. The supersmall sumset property, the \(\mu _G^{(k)}\) and the \(\nu _G^{(k)}\) functions. Unif. Distrib. Theory1 (2006), no. 1, 27–44.
47.
go back to reference S. Ramanujan, Question 464. Journal of the Indian Mathematical Society5 (1913), 120. S. Ramanujan, Question 464. Journal of the Indian Mathematical Society5 (1913), 120.
48.
49.
go back to reference A. G. Vosper, Addendum to “The critical pairs of subsets of a group of prime order”. J. London Math. Soc.31 (1956), 280–282.MathSciNetCrossRef A. G. Vosper, Addendum to “The critical pairs of subsets of a group of prime order”. J. London Math. Soc.31 (1956), 280–282.MathSciNetCrossRef
50.
go back to reference W. D. Wallis, A. P. Street, and J. S. Wallis, Combinatorics: room squares, sumfree sets, Hadamard matrices. Lecture Notes in Mathematics, 292, Springer–Verlag, Berlin-New York, 1972. iv+508 pp. W. D. Wallis, A. P. Street, and J. S. Wallis, Combinatorics: room squares, sumfree sets, Hadamard matrices. Lecture Notes in Mathematics, 292, Springer–Verlag, Berlin-New York, 1972. iv+508 pp.
Metadata
Title
A Walk Through Some Newer Parts of Additive Combinatorics
Author
Béla Bajnok
Copyright Year
2024
DOI
https://doi.org/10.1007/978-3-031-62166-6_2

Premium Partner