The Study Of The Representation Functions In Some Algebraic Structures

Let Z be set of all integers.For set A(?)Z and n?Z,let r(A,An)=#{(a,a')?A×A:a+a'=n}.The function r(A,n)is called the additive representation function of the set A.Let k1,k2 be nonzero integers with(k1,k2)=1.For set A(?)Z and n?Z,let rk1,k2(A,n)=#{(a,a')?A×A:k1a+k2a'=n},we call rk1,k2(A,n)the weighted representation function with weight(k1,k2)of the set A.The research of the additive representation function is related to the well known Erdos-Turan conjecture.In 2003,Nathanson proved that Erdos-Turan conjecture does not hold in Z,he constructed a family of arbitrarily sparse bases A(?)Z satisfying r(A,n)=1 for all n?Z.Let K be a finite field of characteristic?2 and G=K×K.In 2004,Haddad and Helou constructed a set B(?)G satisfying B+B=G for which the.number of representations of g?G as a sum b1+b2(b1,b2?B)is bounded by 18.The main work of this thesis is divided into two parts.In the first part,we generalize the results of the additive representation functions of Z,and obtain the following result:Let f:Z?N0?{?} be a function such that ?=#{f-1(0)} is finite.Let k1,k2 be nonzero integers with(k1,k2)=1 and k1k2?-1.We prove that there exist uncountably many sets A(?)Z such that rk1,k2(A,n)=f(n)for all n?Z.In the second part,we generalize the results of Haddad and Helou in a variety of additive groups derived from finite fields of characteristic?2,and obtain the following result:Let K be a finite field of characteristic?2 and G=K×K.Let k1,k2 be integers not divisible by the characteristic p of K with(k1,k2)=1.For g?G and B(?)G,let rk1,k2(B,9)be the number of solutions of the equation g=k1b1+k2b2,with b1,b2?B.We prove that there exists a set B(?)G such that k1B+k2B=G and rk1,k2(B,g)?16.