C. Covering Systems
(My conjecture on
A system of residue classes is said to be a
(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.
Paul Erdos (1913-1996),
Stefan Znam (1936-1993),
Bernhard Hermann Neumann (1909-2002).
S. Restricted Sumsets
(Linear extension of the Erdos-Heilbronn conjecture
For finite subsets
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
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.
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
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
Sumsets (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."
"While in the past many of the basic combinatorial results were
mainly by ingenuity and detailed reasoning, the modern theory has
out of this early stage, and often relies on deep, well developed tools."
"I am mostly interested in easily stated problems in number theory,
though sometimes this forces me to branch out into combinatorics
and arithmetic geometry."
"Theorems are fun especially when you are the prover, but then the
pleasure fades. What keeps us going are the unsolved problems."
"To pose good unsolved problems is a difficult art."
----Richard K. Guy
Links to Some Homepages|
Electronic J. Combin. Number Theory,
of Number Theory Topics,
World Combinatorics Exchange,
A. S. Fraenkel,
W. T. Gowers,
R. L. Graham,
V. F. Lev,
M. B. Nathanson,
I. Z. Ruzsa,
R. J. Simpson,
R. P. Stanley,
V. H. Vu,
NOTICE: Suggestions and new information are welcome!
Zhi-Wei Sun owns the copyright of this webpage.
The material can only be used for legal academic purposes.