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Open Problems About Sumsets in Finite Abelian Groups: Minimum Sizes and Critical Numbers

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

Abstract

For a positive integer h and a subset A of a given finite abelian group, we let hA, \(h \hat{\;} A\), and \(h_{\pm }A\) denote the h-fold sumset, restricted sumset, and signed sumset of A, respectively. Here we review some of what is known and not yet known about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct and inverse problems.

Keywords

  • Additive combinatorics
  • Finite abelian group
  • Sumset
  • Critical number

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  • DOI: 10.1007/978-3-319-68032-3_2
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Notes

  1. 1.

    Note that \(\lfloor 2 \sqrt{n-2} \rfloor +1=n/p+p\) in this case.

References

  1. N. Alon, M.B. Nathanson, I. Ruzsa, Adding distinct congruence classes modulo a prime. Am. Math. Monthly 102, 250–255 (1995)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. N. Alon, M.B. Nathanson, I. Ruzsa, The polynomial method and restricted sums of congruence classes. J. Number Theory 56, 404–417 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. B. Bajnok, On the maximum size of a \((k, l)\)-sum-free subset of an abelian group. Int. J. Number Theory 5(6), 953–971 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. B. Bajnok, On the minimum size of restricted sumsets in cyclic groups. Acta Math. Hungar. 148(1), 228–256 (2016)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. B. Bajnok, The \(h\)-critical number of finite abelian groups. Unif. Distrib. Theory 10(2), 93–115 (2015)

    MathSciNet  MATH  Google Scholar 

  6. B. Bajnok, More on the \(h\)-critical numbers of finite abelian groups. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 59, 113–122 (2016)

    MathSciNet  Google Scholar 

  7. B. Bajnok, Corrigendum to “The \(h\)-critical number of finite abelian groups”. Unif. Distrib. Theory (to appear)

    Google Scholar 

  8. B. Bajnok, S. Edwards, On two questions about restricted sumsets in finite abelian groups. Australas. J. Comb. 68(2), 229–244 (2017)

    MathSciNet  MATH  Google Scholar 

  9. B. Bajnok, R. Matzke, The minimum size of signed sumsets. Electron. J. Comb. 22(2), paper 2.50, 17 (2015)

    Google Scholar 

  10. B. Bajnok, R. Matzke, On the minimum size of signed sumsets in elementary abelian groups. J. Number Theory 159, 384–401 (2016)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. É. Balandraud, An addition theorem and maximal zero-sum free sets in \(\mathbb{Z}/p\mathbb{Z}\). Israel J. Math. 188, 405–429 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. É. Balandraud, Erratum to: “An addition theorem and maximal zero-sum free sets in \(\mathbb{Z}/p\mathbb{Z}\)”. Israel J. Math. 192(2), 1009–1010 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. É. Balandraud, Addition theorems in \(\mathbb{F}_p\) via the polynomial method, arXiv:1702.06419v1 (math.CO)

  14. A.-L. Cauchy, Recherches sur les nombres. J. École Polytech. 9, 99–123 (1813)

    Google Scholar 

  15. P.H. Diananda, H.P. Yap, Maximal sum-free sets of elements of finite groups. Proc. Jpn. Acad. 45, 1–5 (1969)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. J.A. Dias Da Silva, Y.O. Hamidoune, Cyclic space for Grassmann derivatives and additive theory. Bull. London Math. Soc. 26, 140–146 (1994)

    Google Scholar 

  17. G.T. Diderrich, An addition theorem for abelian groups of order \(pq\). J. Number Theory 7, 33–48 (1975)

    MathSciNet  CrossRef  MATH  Google Scholar 

  18. G.T. Diderrich, H.B. Mann, Combinatorial problems in finite abelian groups. in A Survey of Combinatorial Theory, ed. by J.N. Srivastava et al. (North-Holland 1973)

    Google Scholar 

  19. S. Eliahou, M. Kervaire, Sumsets in vector spaces over finite fields. J. Number Theory 71, 12–39 (1998)

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. S. Eliahou, M. Kervaire, Old and new formulas for the Hopf-Stiefel and related functions. Expo. Math. 23(2), 127–145 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. P. Erdős, H. Heilbronn, On the addition of residue classes mod \(p\). Acta Arith. 9, 149–159 (1964)

    MathSciNet  CrossRef  MATH  Google Scholar 

  22. M. Freeze, W. Gao, A. Geroldinger, The critical number of finite abelian groups. J. Number Theory 129, 2766–2777 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  23. M. Freeze, W. Gao, A. Geroldinger, Coorigendum to “The critical number of finite abelian groups”. J. Number Theory 152, 205–207 (2015)

    MathSciNet  CrossRef  MATH  Google Scholar 

  24. L. Gallardo, G. Grekos et al., Restricted addition in \(\mathbb{Z}_ /n\mathbb{Z}\) and an application to the Erdős–Ginzburg–Ziv problem. J. London Math. Soc. 65(2), 513–523 (2002)

    Google Scholar 

  25. W. Gao, Y.O. Hamidoune, On additive bases. Acta Arith. 88(3), 233–237 (1999)

    MathSciNet  CrossRef  MATH  Google Scholar 

  26. J.R. Griggs, Spanning subset sums for finite abelian groups. Discret. Math. 229, 89–99 (2001)

    MathSciNet  CrossRef  MATH  Google Scholar 

  27. Y.O. Hamidoune, A. Plagne, A new critical pair theorem applied to sum-free sets in Abelian groups. Comment. Math. Helv. 79, 1–25 (2003)

    Google Scholar 

  28. Gy. Károlyi, On restricted set addition in abelian groups. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 46, 47–54 (2003)

    Google Scholar 

  29. Gy. Károlyi, The Erdős–Heilbronn problem in abelian groups. Israel J. of Math. 139, 349–359 (2004)

    Google Scholar 

  30. Gy. Károlyi, A note on the Hopf–Stiefel function. Eur. J. Combin. 27, 1135–1137 (2006)

    Google Scholar 

  31. J.H.B. Kemperman, On small sumsets in an abelian group. Acta Math. 103, 63–88 (1960)

    MathSciNet  CrossRef  MATH  Google Scholar 

  32. V.F. Lev, Restricted set addition in groups I: the classical setting. J. London Math. Soc.62(2), 27–40 (2000)

    Google Scholar 

  33. V.F. Lev, Three-fold Restricted Set Addition in Groups. Europ. J. Combinatorics 23, 613–617 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  34. V.F. Lev, Critical pairs in abelian groups and Kemperman’s structure theorem. Int. J. Number Theory 3, 379–396 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  35. H.B. Mann, Y.F. Wou, Addition theorem for the elementary abelian group of type \((p, p)\). Monatshefte für Math. 102, 273–308 (1986)

    MathSciNet  CrossRef  MATH  Google Scholar 

  36. N.H. Nguyen, E. Szemerédi, V.H. Vu, Subset sums modulo a prime. Acta Arith. 131(4), 303–316 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  37. A. Plagne, Additive number theory sheds extra light on the Hopf–Stiefel \(\circ \) function. Enseign. Math., II Sér. 49 (1–2), 109–116 (2003)

    Google Scholar 

  38. A. Plagne, Optimally small sumsets in groups, I. The supersmall sumset property, the \(\mu _G^{(k)}\) and the \(\nu _G^{(k)}\) functions. Unif. Distrib. Theory 1(1), 27–44 (2006)

    MathSciNet  MATH  Google Scholar 

  39. A. Plagne, Optimally small sumsets in groups, II. The hypersmall sumset property and restricted addition. Unif. Distrib. Theory 1(1), 111–124 (2006)

    MathSciNet  MATH  Google Scholar 

  40. D. Shapiro, Products of sums of squares. Expo. Math. 2, 235–261 (1984)

    MathSciNet  MATH  Google Scholar 

  41. A.G. Vosper, The critical pairs of subsets of a group of prime order. J. Lond. Math. Soc. 31, 200–205 (1956)

    MathSciNet  CrossRef  MATH  Google Scholar 

  42. A.G. Vosper, Addendum to "The critical pairs of subsets of a group of prime order". J. Lond. Math. Soc. 31, 280–282 (1956)

    MathSciNet  CrossRef  MATH  Google Scholar 

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Bajnok, B. (2017). Open Problems About Sumsets in Finite Abelian Groups: Minimum Sizes and Critical Numbers. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_2

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