Combinatorial Number Theory in China
Zhi-Wei Sun
Nanjing University
Nanjing 210093, P. R. China
zwsun@nju.edu.cn
http://math.nju.edu.cn/∼zwsun

Nov. 6, 2009

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, andoften relies on deep, well developed tools.
—Noga Alon (ICM, Beijing, 2002)
Additive combinatorics is currently a highly active area ofresearch. One remarkable feature of the field is the use oftools from many diverse fields of mathematics.
—Terence Tao & V. H. Vu (2006)

What is Combinatorial Number Theory ?
Combinatorial Number Theory studies number-theoreticproblems of combinatorial favor or the combinatorics of integers.
Topics in CNT include: Additive combinatorics, Ramsey theoryof integers, combinatorial congruences (not combinatorialidentities), etc.
Tools: Elementary method, algebraic method, analytic method,and mixture of several methods.
Two typical Problems in CNT:
1. (Olson) Is it true that for any a1 , . . . , ak(n−1)+1 ∈ Zkn (the direct
sum of k copies of the cyclic group Zn = Z/nZ) we can selectsome of them so that their sum is zero?
2. (Snevily’s Conjecture) Let G be any abelian group of odd order.
Then, for any subsets A and B of G with cardinality k we canwrite A = {a1 , . . . , ak } and B = {b1 , . . . , bk } with
a1 + b1 , . . . , ak + bk pairwise distinct.

Leading Mathematicians in CNT
Paul ErdősRonald Graham
Noga Alon
E. Szemeredi
W. Gowers
Terence TaoBen GreenI. Z. Ruzsa
M. B. Nathanson
A. SárközyCarl PomeranceThree main CNT teams in China
Wei-Dong Gao’s School: Study zero-sum sequences for abeliangroups. [I include Prof. P. Z. Yuan in Gao’s school since he was
led to the field of zero-sums by Prof. Gao.]
Remark. Gao and his students view them as combinatorists.
Yong-Gao Chen’s School: Study sumset problems andapplications of covers.
Remark. Chen and his students view themselves as number
theorists.
Zhi-Wei Sun’s School: Study covers of the integers or groups,
restricted sumsets via Combinatorial Nullstellensatz, and various
combinatorial congruences.
Remark. Sun and his students view themselves as combinatorists
and number theorists alternatively.

Representative work of W. D. Gao
Gao’s Theorem [conjectured by Y. Caro]. Let G be a finite
additive abelian group. Let D(G ) (the Davenport constant for G )[respectively, s(G )] be the smallest positive integer n such that for
any a1 , . . . , an ∈ G there are distinct i1 , . . . , ik ∈ {1, . . . , n} [resp.,with k = |G |] satisfying ai1 + · · · + aik = 0.Thens(G ) = D(G ) + |G | − 1.
References
1. W. D. Gao, A combinatorial problem on finite abelian groups, J.
Number Theory 58(1996), 100–103.
2. M. Devos, Gao’s theorem for nonabelian groups, Open Problem
Garden, http://garden.irmacs.sfu.ca/?q=op/gaos theorem for nonabelian groups
Conjecture: Gao’s theorem holds for any finite group G .

Representative work of P. Z. Yuan
Let G be a finite abelian group and A a finite subset of Z. Define
DA (G ) (resp., EA (G )) to be the smallest positive integer n suchthat for any x1 , . . . , xn ∈ G there are a1 , . . . , ak ∈ P
A (resp., withk = |G |) and distinct j1 , . . . , jk ∈ {1, . . . , n} with ki=1 ai xji = 0.
Adhikari-Thangadurai Conjecture. EA (G ) = DA (G ) + |G | − 1.
A Result of P. Z. Yuan and X. Zeng (2009). The conjectureholds for all finite cyclic groups.
Inspired by Yuan and Zeng’s preprint, D. J. Grynkiewicz et al.
proved the whole conjecture.
References
1. P. Yuan and X. Zeng, Davenport constant with weights,preprint, 2009.
2. D. J. Grynkiewicz, L. E. Marchan and O. Ordaz, A weighted
generalization of two theorems of Gao, preprint,
arXiv:0903.2810.

Representative work of Y. G. Chen
For n = 1, 2, 3, . . . defineg (n) = max{|S| : S ⊆ Z+ and [x, y ] 6 n for all x, y ∈ S}.
In 1951 P. Erdős asked for values of g (n).
In 1998 Y. G. Chen successfully provided an asymptotic formula.
q√
Chen’s Theorem. g (n) = 98 n + o( n).
Let A be an infinite set of positive integers. In 1962 P. Erdős√conjectured that if A(n) =P|{a ∈ A : a 6 n}| > c n (where c is apositive constant), then { a∈S a : S ⊆ A} contains an infinite
arithmetic progression.
Y. G. Chen gave the first proof of this conjecture.
References
1. Y. G. Chen, Sequences with bounded l.c.m. of each pair ofterms, Acta Arith. 84(1998), 71–95.
2. Y. G. Chen, On subset sums of a fixed set, Acta Arith.
106(2003), 207–211.

Representative work of Z. W. Sun
A main contribution of Sun to CNT is the discovery of connections
between zero-sum problems for abelian groups and covers of theintegers by residue classes.
Theorem [Z. W. Sun, Israel J. Math. 2009]. Let
A = {a1 (mod n1 ), . . . , ak (mod nk )}be a finite collection of residue classes, and let G be any abelianp-group. If A covers each integer either (exactly) 2|G | − 1 or 2|G |times, thenPfor any c1 , . . . , ck ∈ P
G there exists an I ⊆ {1, . . . , k}such that i∈I 1/ni = |G | and i∈I ci = 0.
Remark. It is interesting to view 1/ni as a weight of i. In the casen1 = · · · = nk = 1, the result of Sun yields the classicalErdős-Ginzburg-Ziv theorem. By a recent paper of H. Pan and L.
L. Zhao [Adv. in Appl. Math. 2009], for each m = 2, 3, . . . there
are infinitely many exact m-covers which cannot split into twocovers of Z.

Representative work of Z. W. Sun (continued)
In the direction of Snevily’s conjecture, Sun established a3-dimensional analogue of Snevily’s conjecture.
Theorem [Z. W. Sun, Math. Res. Lett. 2008] Let G be any
additive abelian group with cyclic torsion subgroup, and let A, B
and C be finite subsets of G with cardinality n > 0. Then there isa numbering {ai }ni=1 of the elements of A, a numbering {bi }ni=1 ofthe elements of B and a numbering {ci }ni=1 of the elements of C ,such that all the sums ai + bi + ci (1 6 i 6 n) are (pairwise)
distinct. Consequently, each subcube of the Latin cube formed by
the Cayley addition table of Z/NZ contains a Latin transversal.
Remark. We don’t require that |G | is odd. The theorem does nothold for general abelian groups. It even fails for the Klein groupZ2 ⊕ Z2 .

Representative work of Z. W. Sun (continued)
In the direction of covers of groups by cosets, Sun made animportant progress on the Herzog-Schönheim conjecture.
Theorem [Z. W. Sun, J. Algebra 2004] Let A = {ai Gi }ki=1 be a
finite system of left cosets in a group G with all the Gi subnormal
in G but not all equal to G . Suppose that A covers all theelements of G the same number of times, and that among theindicesn1 = [G : G1 ] 6 . . . 6 nk = [G : Gk ].
each occurs at most M ∈ Z+ times. Then M > 1, i.e,
[G : Gi ] = [G : Gj ] for some 1 6 i < j 6 k. Furthermore,
log n1 6eγ
M log2 M + O(M log M log log M),
log 2

where γ is Euler’s constant and the O-constant is absolute.
Remark. A subgroup H of G is called subnormal if there is a chainof subgroups H = H0 ⊆ H1 ⊆ · · · ⊆ Hn = G such that Hi isnormal in Hi+1 for all i = 0, . . . , n − 1.

Representative work of Z. W. Sun (continued)

References
1. Z. W. Sun, Unification of zero-sum problems, subset sums andcovers of Z, Electron. Res. Announ. Amer. Math. Soc. 9(2003),
51–60.
2. Z. W. Sun, Zero-sum problems for abelian p-groups and coversof the integers by residue classes, Israel. J. Math. 170(2009),
235–252.
3. Z. W. Sun, An additive theorem and restricted sumsets, Math.
Res. Lett. 15(2008), 1263–1276.
4. Z. W. Sun, On the Herzog-Schönheim conjecture for uniform
covers of groups, J. Algebra 273(2004), 153–175.

Some conjectures of Z. W. Sunk
Conjecture 1 (Sun, 2003) If A
P= {ai (mod ni )}i=1 covers every
integer at least m times then i∈I 1/ni ∈ mZ for some∅=6 I ⊆ {1, . . . , k}.

Conjecture 2 (Sun, 2004) If a1 G1 , . . . , ak Gk (k > 1) are disjoint
left cosets in a group G with [G : Gi ] < ∞ for all i = 1, . . . , k,then gcd([G : Gi ], [G : Gj ]) > k for some 1 6 i < j 6 k.
Conjecture 3 (Sun, 2008). Let a1 , . . . , an be nonzero elements ofa field F . Then, for any finite A ⊆ F we have|{a1 x1 + · · · + an xn : x1 , . . . , xn are distinct elements of A}|> min{p(F ) − δ, n(|A| − n) + 1},
where δ = 1 if n = 2 and a1 + a2 = 0, and δ = 0 otherwise.
Conjecture 4 (Sun, 2008). Any n × n × n Latin cube contains aLatin transversal.
Conjecture 5 (Sun, 2009) Let r (n) be the number of ways towrite n in the form x 2 + (y 2 − 3y )/2 + (2z 2 − z) with x, y , z ∈ N.
Then {r (n) : n ∈ Z+ } = Z+ .

The second generation
W. D. Gao, Y. G. Chen and Z. W. Sun can be viewed as therepresentatives of the first generation of combinatorial number
theorists in China. They taught CNT by themselves. Gao is goodat the tool of group-rings, Chen is good at analytic method, andSun is familiar with algebraic methods and combinatorics. Now
they are all over forties.
The second generation of Chinese combinatorial number theorists
consists some young professors (or associate prof.) with ages
between 30-37. Below is a list of main representatives of thesecond generation.
Gao’s School: Ju-Juan Zhuang (Dalian Martime Univ.), et al.
Chen’s School: Li-Xia Dai (Nanjing Normal Univ.), Min Tang(Anhui Normal Univ.), Shu-Guang Guo (Yancheng NormalCollege), Xue-Gong Sun (Huaihai Institute of Technology), et al.
Sun’s School: Hao Pan (Nanjing Univ.), Hui-Qin Cao (Nanjing
Auditing College), Song Guo (Huaiyin Normal College), et al.

The third generation

The third generation

The third generation is among
the graduate students
participating this conference.

Thank you!