The Study On The Analogous Of Erd(?)s-Turán Conjecture In Two Algebraic Structures

In 1941, Erdo(o|¨)s and Turán formulated the famous Erdo(o|¨)s-Turán conjecture when they studied the additive representation functions.This conjecture have had an important impact in the field of additiverepresentation functions. Mathematicians carried out a lot of studies,such as the mean values of representation functions, the analogousof Erdo(o|¨)s-Turán conjecture in some algebraic structures, and so on.These works promoted the development of additive number theorylargely.Let S be a semigroup. For A(?)S and g∈S, we defineδA(g) =#{(a,a' )∈A×A : a - a' = g}. In this dissertation, we investigatethe analogous of Erdo(o|¨)s-Turán conjecture in two algebraic structures.The main results are summarized as follows.1. Let K be a finite field of characteristic (?) 2 and G the additivegroup of K×K, then there exists a set B (?) G such that B ?B = G,andδB(g)≤14 for all g (?) 0. This result means that the analogous ofErdo(o|¨)s-Turán conjecture does not hold in a variety of additive groupsderived from certain fields.2. There exists a family of set A (?) Z such that A (?) A = Z andδA(n) = 1 for all n(= 0)∈Z. Moreover, we construct a uniquedifference basis A of Z such that2log(3x+3)/log3 - 2log 5/log 3< A(0, x)≤2log(x+3)/log2-2.where A(0, x) = #{a∈A : 0≤a≤x}. This result shows that theanalogue of Erdo(o|¨)s-Turán conjecture does not hold in (Z, -).