Some Problems In Combinatorial Number Theory

Both subset sums and zero-sum problems are important topics in combinatorial number theory.In this thesis,we study three problems related to them:restricted sumsets, Davenport constant and short zero-sum subsequences.This thesis consists of four chapters.ChapterⅠcontains some basic notations and concepts.In Chapter 2 we give a natural generalization of Davenport constant to any finite commutative semigroup.Let G be a finite commutative semigroup.The Davenport constant D(G) of G is the smallest integer e such that,every sequence S of e elements in G contains a subsequence T(≠S) with the same product of S.Let R=Z_n1?…?Z_nr. Among other results,we determine D(R^×)-D(U(R)),where R^×is the multiplicative semigroup of R and U(R) is the group of units of R.In Chapter 3 we focus on the restricted sumsets in abelian groups.Let G be an abelian group of odd order,and let A be a subset of G.For any integer h such that 2≤h≤|A|-2,we prove that |h^A|≥|A| and the equality holds if and only if A is a coset of some subgroup of G,where h^A is the set of all sums of h distinct elements of A.In Chapter 4 we investigate some problems concerned with short zero-sum subsequences. Let exp(G) be the exponent of a finite abelian group G.We call a zero-sum sequence T in G a short zero-sum sequence if 1≤|T|≤exp(G).Let S be a sequence of elements in G,and let h(S) be the maximal repetition of S.We obtain some nontrivial lower bound of|∑(S)| in case when S contains no zero-sum subsequence of length in [1,h(S)].We also prove that for some type of finite abelian groups G,there exists some integer t of[exp(G)+1,ρ(G)-1]such that every zero-sum sequence S of elements in G with length |S|=t contains a short zero-sum subsequence,whereρ(G) is the smallest integer e such that every sequence in G with length at least e contains a short zero-sum subsequence.