Projections, entropy and sumsets

Abstract

In this paper we shall generalize Shearer’s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    G.R. Allan: An inequality involving product measures, in: Radical Banach Algebras and Automatic Continuity (J.M. Bachar et al., eds.), Lecture Notes in Mathematics 975, Springer-Verlag, 1981, 277–279.

  2. [2]

    N. Alon and M. Dubiner: Zero-sum sets of prescribed size, in: Combinatorics, Paul Erdős is eighty, Vol. 1, pp. 33–50, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993.

    Google Scholar 

  3. [3]

    P. Balister and J.P. Wheeler: The Erdős-Heilbronn problem for finite groups, Acta Arithmetica, 140 (2009), 105–118.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    B. Bollobás and A. Thomason: Projections of bodies and hereditary properties of hypergraphs, Bull. London Math. Soc. 27 (1995), 417–424.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    Yu.D. Burago and V.A. Zalgaller: Geometric Inequalities, Springer-Verlag, 1988, xiv+331pp.

  6. [6]

    F.R.K. Chung, R.L. Graham, P. Frankl and J.B. Shearer: Some intersection theorems for ordered sets and graphs, J. Combinatorial Theory A 43 (1986), 23–37.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    H. Davenport: On the addition of residue classes, J. London Math. Soc. 10 (1935), 30–32.

    Article  Google Scholar 

  8. [8]

    A. Frank: Edge-connection of graphs, digraphs, and hypergraphs, in: More Sets, Graphs and Numbers, Bolyai Soc. Math. Stud. 15, Springer, Berlin, 2006, pp. 93–141.

    Google Scholar 

  9. [9]

    K. Gyarmati, M. Matolcsi and I. Z. Ruzsa: Plünnecke's inequality for different summands, in: Building Bridges, Bolyai Soc. Mathematical Studies 19, ed. M. Grötschel, G. O. H. Katona, Springer-Bolyai 2008, 309–320.

  10. [10]

    K. Gyarmati, M. Matolcsi and I. Ruzsa: A superadditivity and submultiplicativity property for cardinalities of sumsets, Combinatorica 30 (2010), 163–174.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    H. Hadwiger: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, 1957, xiii+312pp.

  12. [12]

    T. S. Han: Nonnegative entropy measures of multivariate symmetric correlations, Information and Control 36 (1978), 133–156.

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    G. Károlyi: The Cauchy-Davenport theorem in group extensions, L'Enseignement Mathématique 51 (2005), 239–254.

    MATH  Google Scholar 

  14. [14]

    L.H. Loomis and H. Whitney: An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961–962.

    MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    L. Lovász: Solution to Problem 11, see pp. 168–169 of’ Report on the 1968 Miklós Schweitzer Memorial Mathematical Competition’ (in Hungarian), Matematikai Lapok 20 (1969), 145–171.

    Google Scholar 

  16. [16]

    L. Lovász: 2-matchings and 2-covers of hypergraphs, Acta Math. Acad. Sci. Hungar. 26 (1975), 433–444.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    L. Lovász: On two minimax theorems in graph, J. Combinatorial Theory Ser. B 21 (1976), 96–103.

    MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    M. Madiman and P. Tetali: Sandwich bounds for joint entropy, Proc. IEEE Intl Symp. Inform. Theory, Nice, June, 2007.

  19. [19]

    M. Madiman and P. Tetali: Information inequalities for joint distributions, with interpretations and applications, IEEE Transactions on Information Theory 56 (2010), 2699–2713.

    MathSciNet  Article  Google Scholar 

  20. [20]

    I. Z. Ruzsa: Cardinality questions about sumsets, in: Additive Combinatorics, CRM Proc. Lecture Notes 43, Amer. Math. Soc., Providence, RI, 2007, pp. 195–205.

    Google Scholar 

  21. [21]

    J. P. Wheeler: The Cauchy-Davenport theorem for finite groups, preprint, http://www.msci.memphis.edu/preprint.html (2006).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Béla Bollobás.

Additional information

Research supported in part by NSF grants CCR-0225610, DMS-0505550 and W911NF-06-1-0076

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Balister, P., Bollobás, B. Projections, entropy and sumsets. Combinatorica 32, 125–141 (2012). https://doi.org/10.1007/s00493-012-2453-1

Download citation

Mathematics Subject Classification (2000)

  • 52B60
  • 11P99