Covering Systems, Restricted Sumsets,
Zero-sum Problems and their Unification
Initial day: May 4, 2003      Last modified: 2009-07-07
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C. Covering Systems (My conjecture on disjoint cosets NEW)

      A system of residue classes is said to be a covering system
    (or a cover of Z in short) if each integer lies in at least one
    of the residue classes. The main problem is to investigate the
    moduli in a cover of Z.

     Letters from Paul Erdos (1913-1996), Stefan Znam (1936-1993),
             and Bernhard Hermann Neumann (1909-2002).

S. Restricted Sumsets (Linear extension of the Erdos-Heilbronn conjecture NEW)

      For finite subsets A1,...,An of a field F and a polynomial
    f(x1,...,xn) over F, we want to know sharp lower bound of the
    cardinality of the restricted sumset
    {a1+...+an: a1Î A1,...,anÎ An, f(a1,...,an)¹0}
    in terms of |A1|,...,|An|.

Z. Zero-sum Problems (Review of C. Reiher's solution to the Kemnitz conjecture)

      Let G be a finite additive abelian group. The zero-sum problem on
    G asks for the least positive integer k such that for any k elements
    of G (repetition allowed) we can select a given number of them whose
    sum equals zero.

     NEWS: D. J. Grynkiewicz recently proved the weighted EGZ theorem!

U. Unification of the Three Topics (with the published version in Israel J. Math.)

      The above three topics initiated by the great mathematician P. Erdos
    play important roles in both number theory and combinatorics, they have
    attracted lots of researchers and many surprising applications have been
    found. However, only in 2003 Zhi-Wei Sun revealed connections among these
    seemingly unrelated topics and presented a unified approach for the first

     Open Conjectures on the Topics      My Main Work on the Topics (Z. W. Sun)
     V. F. Lev's Problems Page      Covers of Z with k<10 Residue Classes

Related Surveys

    C1. S. Porubsky and J. Schonheim, Covering systems of Paul Erdos: past, present and future,
      in: Paul Erdos and his Mathematics. I (edited by G. Halasz, L. Lovasz, M. Simonvits,
      V. T. Sos), Bolyai Soc. Math. Studies 11, Budapest, 2002, pp. 581--627. MR 2004d:11006.

    C2. Z. W. Sun, Recent Progress on Covers of the Integers and Their Applications, 2000.
    C3. Z. W. Sun, Equalities and Inequalities Related to Covers of Z or Groups, 2002.
    C4. Z. W. Sun, On the Structure of Periodic Arithmetical Maps, 2002.
    C5. Z. W. Sun, Problems and Results in Combinatorial Number Theory, 2002.
    C6. S. Porubsky, Results and Problems on Covering Systems of Residue Classes,
      Mitt. Math. Sem. Giessen, Heft 150, Giessen Univ., 1981, pp.1--85.
    C7. S. Znam, A Survey of Covering Systems of Congruences,
      Acta Math. Univ. Comenianae, 40/41(1982), 59-79.
    C8. Z. W. Sun, Self-introduction to Sun's Papers on Covers, updated.
    C9. Z. W. Sun, Two Local-Global Theorems and a Powerful Formula, 2004.
   C10. Z. W. Sun, Problems and Results on Covering Systems, 2005.
    S1. N. Alon, Discrete mathematics: methods and challenges,
      in: Proceedings of the International Congress of Mathematicians
      (Beijing, 2002), Vol. I, Higher Education Press, Beijing, 2003, 119--135.
    S2. N. Alon,
Combinatorial Nullstellensatz, Combin. Probab. Comput. 8(1999), 7--29.
    S3. Z. W. Sun,
New Results on Subset Sums, 2002.
    S4. Z. W. Sun, Sumsets with Polynomial Restrictions, 2002.
    S5. Z. W. Sun, On Some Conjectures of Erdos-Heilbronn, Lev and Snevily, 2004, 2005.
    S6. Z. W. Sun, Additive Combinatorics and Latin Transversal, 2007.
    S7. E. Szemeredi and V. H. Vu, Long Arithmetic Progressions in Sum-sets
      and the Number of $x$-sum-free Sets, 2003.
    S8. T. Tao, Some Highlights of Arithmetic Combinatorics, 2003.
    Z1. Y. Caro, Zero-sum Problems--a Survey, Discrete Math. 152(1996), 93--113.
    Z2. Z. W. Sun, On Zero-sum Problems, 2002.
    CZ. Z. W. Sun, Covering Systems and their Connections to Zero-sums, 2004.
   SZ1. M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of
(Graduate texts in math.; 165), Springer-Verlag, New York, 1996.
   SZ2. T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006.
   CSZ1. Z. W. Sun, Recent Progress on Combinatorial Number Theory (Chinese), 2001.
   CSZ2. R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition,
      Springer-Verlag, 2004, sections A19, B21, C8, C15, E23, F13, F14.
   CSZ3. Z. W. Sun, Groups and Combinatorial Number Theory, Oct. 2004.


Selected Words from Experts

    "Perhaps my favourite problem of all concerns covering systems."
                           ----Paul Erdos

    "While in the past many of the basic combinatorial results were obtained
    mainly by ingenuity and detailed reasoning, the modern theory has grown
    out of this early stage, and often relies on deep, well developed tools."

                           ----Noga Alon

    "I am mostly interested in easily stated problems in number theory,
    though sometimes this forces me to branch out into combinatorics
    and arithmetic geometry."

                           ----Andrew Granville

    "Theorems are fun especially when you are the prover, but then the
    pleasure fades. What keeps us going are the unsolved problems."

                           ----Carl Pomerance

    "To pose good unsolved problems is a difficult art."
                           ----Richard K. Guy

Links to Some Homepages

    Number Theory Web, INTEGERS: Electronic J. Combin. Number Theory,
    List of Number Theory Topics, The World Combinatorics Exchange,
    A. S. Fraenkel, A. Geroldinger, W. T. Gowers, R. L. Graham, B. Green,
    G. Karolyi, V. F. Lev, M. B. Nathanson, S. Porubsky, I. Z. Ruzsa,
    R. J. Simpson, H. Snevily, R. P. Stanley, T. Tao, V. H. Vu, D. Zeilberger.

    NOTICE: Suggestions and new information are welcome!
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    The material can only be used for legal academic purposes.