On An Inverse Problem In Additive Number Theory

In this thesis,we mainly research on the properties of perturbed sequences and an inverse problem in additive number theory.The main results are summarized as follows.1.Let ? be a positive real number and Q?={[?n2]n ? Z+} be the perturbed sequence.Hegyvari proved that,let ? be a positive real number and ?(?)Z+,then:(i)? is of the form 2k+1/2 or 2k+2/3,where k is a nonnegative integer,if and only if every element of Q? is even;(ii)? is of the form 3k+3.where k is a nonnegative integer,if and only if every even;(ii)a is of the form 3k+3/4,where is a nonnegative integer,if and only if every element of Qa is divisible by 3.In this thesis,we consider the perturbed sequence P?=(?)and prove that(Journal of Mathematics in Practice and Theory(China),48(2018),297-301),let ? be a positive real number and a(?)Z+,then:(i)? is of the form 2k+2/3 or 2k+4/5,where k is a nonnegative integer,if and only if every element in P? are even.(ii)For any given prime p?3,there is no a such that every element in P? is divisible by p.2.Let N be the set of all natural numbers.For a sequence of positive integers A,let P(A)be the set of all integers which can be represented as the finite sum of distinct terms of A.In 2012,by improving Hegyvari's result,Chen and Fang proved that:(i)Let B={b1<b2<…} be a sequence of integers with b1? {4,7,8}U {b:b? 11,b? N} and bn+1? 3bn+5 for all n?1.Then there exists a sequence of positive integers A for which P(A)=N\B.(ii)Let B={b1<b2<…} be a sequence of integers with b2=3b1+4.Then there is no sequence A of positive integers for which P(A)=N\B.In this thesis,we prove that(Acta Math.Hungar.158(2019),36-39):if A and B={1<b1<b2<…} are two infinite sequences of positive integers with b2=3b1+5 such that P(A)=N\B,then b3? 4b1+6.Furthermore,there exist two infinite sequences of positive integers A and B={1<b1<b2<…} with b2=3b1+5,b3=4b1+6 such that P(A)=N\B.