On Zero-sum And Arithmetic Problems In Combinatorial Number Theory

In this thesis,we mainly study several important problems in combinatorial number theory:the direct and inverse problems of the invariant disc(G),the direct and inverse problems associated with invariant skexp(G)(G),the problem of the greatest common divisor of certain binomial coefficients,and the upper bound problem for the sum of reciprocals of least common multiples.In Chapter 2,let G be an additive finite abelian group.By disc(G)we denote the smallest positive integer t such that every sequence S over G of length |S|?t has two nonempty zero-sum subsequences of distinct lengths.In Chapter 2,we first extend the list of the groups G for which disc(G)is known.Then we focus on the inverse problem associated with disc(G).Let L1(G)denote the set of all positive integers t with the property that there is a sequence S over G with length |S|=disc(G)-1 such that all nonempty zero-sum subsequences of S have the same length t.We determine L1{G)for some special groups including the groups with large exponents compare to |G|/exp(G),the groups of rank at most two,the groups Cpnr with 3 ? r?p and the groups Cmpn?H.where H is a p-group with D(H)?pn,and D(H)denotes the Davenport constant of H.In particular,we find some groups G with |L1(G)|?2 which disproves a conjecture suggested by Gao,Li,Zhao and Zhuang.Let S be a sequence over G such that all nonempty zero-sum subsequences have the same length.We determine the structure of S for the cyclic group Cn when |S|?n+1,and for the group Cn ? Cn when |S|=3n-2=disc(Cn ? Cn)-1.In Chapter 3,let G be an additive finite abelian group of exponent exp(G).For every positive integer k,let skexp(G)(G) denote the smallest integert such that ev-ery sequence over G of length t has a zero-sum subsequence of length kexp(G).Let ?kexp(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length between 1 and kexp(G).It is conjectured by Gao et al.that skexp(G)(G)=?kexp(G)(G)+kexp(G)-1 for all pairs of(G,k).This conjecture is a common generalization of several previous conjectures and has been confirmed for some special pairs of(G,k).In Chapter 3,we shall prove this conJ ecture for more pairs of(G,k).We also study the inverse problem associated with skexp(G)(G),i.e.,determin-ing the sequence S of length skexp(G)(G)-1 such that S has no zero-sum subsequence of length kexp(G).In Chapter 4,let m and n be positive integers.Let((?))=m!/n!(m-n)! stand for the binomial coefficient indexed by m and n,where n! is the factorial of n.For any prime 7.let vp(n) denote the largest nonnegative integer r such that pr divides n.In Chapter 4,we use the p-adic method to show the following identity:(?).This extends greatly the identities obtained by Mendelsoh et al in 1971 and by Albree in 1972,respectively.In Chapter 5,let n and k be positive integers such that n?k+1 and let {ai}i=1n be an arbitrary given strictly increasing sequence of positive integers.Let (?).In 1978,Borwein proved a conjecture of Erdos stating that if n?2,then Sn,1?1-1/2n-1,with the equality holding if and only if ai=2i-1 for l<i<n.In Chapter 5,we first improve Borwein's upper bound by showing that Sn,1?1/a1(1-1/2n-1)with the equality occurring if and only if ai=2i-1a1 for all integers i with 1?i?n.Then we use this improved upper bound to show that if n>3,then Sn,2?7/6+1/2|n/2|(2/3?n-7/3),with the equality holding if and only if a1=1,a2i=2i and a2i+1=3 x 2i-1 for all integers i with 1?i?[n/2].where ?n:=0 if n is even,and 1 if n is odd.Furthermore,we show that if n?7,then Sn,3?17/15-37/15·1/2[n/3]+?n/2[n/3],with the equality occurring if and only if ai=i for all i?{1,2,3} and a3i+1=2i+1(1?i?[n-1/3]),a3i+2=5 × 2i-1(1?i?[n-2/3]])and a3i+3=3 × 2i(1 ?i?[n/3]-1),where ?n=0 if 3|n,if n ? 1(mod 3)and 9/5 if n? 2(mod 3).We also present tight upper bound for Sn.3 if n ? {4,5,6}.