Erdos-Turan Conjecture, Representation Functions And Some Other Additive Problems

In this thesis we investigate the Erdos-Turan conjecture, representation functions, sumsets and difference sets, and additive complements. The main results are as fol-lows.1. The Erdos-Turan conjectureLet N be the set of all nonnegative integers and A(?)N. For n∈N, let σA (n) denote the number of solutions of α1+α2=n,α1,α2∈A. IfσA(n)≥1for all nonnegative integers n,then A is called a basis of N. If σA(n)≥1for all sufficiently large integers n, then A is called an asymptotic basis of N.In1941, Erdos and Turan posed the following famous conjecture:If A is an asymptotic basis of N, then σA(n) is unbounded.In1990, Ruzsa proved that there exists a basis A of N such thatIn2008, Tang proved that there exists a basis A of N such that for all sufficiently large integers N.In2010, Tang proved that there exists a basis A of N such that for all integers N≥7.628517798x1027.In2013, Chen and Yang gave a simple proof of Ruzsa’s result.(Published in European J. Combin.) For any real numberα>1,let Sα(A,N)be the α-th power mean of σA(1),σA(2),…,σA(N).In this thesis,we prove the following results:(i)There exists a basis A of N such that∑n≤Nσ2A(n)≤670N for all integers N≥1.(ii)If A is an asymptotic basis of N,then,for any real number α>0,we have Sα(A,N)≥2+ON(1).(iii)For0<α<1,there exists a basis A of N such that Sα(A,N)=2+ON(1) aNd#{1≤n≤N:σA(n)=2)=N+O(Nlog3/log4).2.Representation functionsFor A(?)N and n∈N,let R1(A,n)=σA(n),and let R2(A,n)and R3(A,n) denote the number of solutions of α+α’=n,α,α’∈A,α<α’and α+α’＝n，α，α’∈A,α≤α’respectively.Fori∈{1,2,3},Sarkozy asked whether there are sets A and B with|(A∪B)\(A∩B)|=+∞such that Ri(A,n)=Ri(B,n)for all sufficiently large integers n.In2002,Dombi solved the cases i=1and i=2.In2003,Chen and Wang solved the case i=3.Sets A and B,which are constructed by Dombi for i=2,Chen and Wang for i=3,satisfy B=N\A.Later,Lev,sandor and Tang gave several different proofs.Suppose that κ1,κ2are integers.Let γκ1,κ2(A,n)denote the number of solutions of the equation n=κ1α1+κ2α2,α1,α2∈A.In this thesis,we prove the following results(Published in J.Number Theory):Theorem1.Let κ1and κ2be two integers with κ2≥κ1≥1.Then a necessary and sufficient condition that there exists a set A(?)N such that γκ1,Κ2(A,n)=γκ1,κ2(N\A,n) holds for all sufficiently large integers n is κ1κ2and κ2>κ1.3. Sumsets and difference setsSuppose that A(?)N. Let A+A and A-A denote the set of all integers of the form α1+α2,α1，α2∈A and α1-α2，α1，α2∈A respectively.In1986, Erdos and Freud posed the following conjecture:If A(?){1,2,..., n} with|A|> n/3, then there exists some power of2which is the sum of elements of A.In1990, Erdos and Freiman proved this conjecture. Taking A={α:α∈{1,..., n},3|α}, we know that the result above is best possible.In2010, Kapoor proved that for an unbounded sequence{ακ} of positive integers with ακ+1/ακ�'αas κ�'∞, and a real number β＞max(α,2), if A(?)[0, x]∩N with0∈A and then A+A contains a term of the sequence{ακ}.In this thesis, we prove the following result (Published in Bull. Aust. Math. Soc):Theorem2. Let β>1be a real number, and let{ακ} be an unbounded sequence of positive integers such that ακ+1/ακ≤βfor all κ≥1. Suppose that n is an integer with n>(1+1/(2β))α1and A is a subset of{0,1,..., n}.(i) If all ακ are even, and then (A+A)∩(A-A) contains a term of{αk};(ii) If αk are not all even, and then (A+A)∩(A-A) contains a term of{αk}. Furthermore (i) and (ii) are sharp. In addition, we have the following corollary.Corollary1. Let n≥3be an integer and A be a subset of{0,1,..., n} such that|A|>4n/5. Then (A+A)∩(A-A) contains a power of2. Furthermore,4/5cannot be improved.4. Additive complementsLet Z denote the set of all integers. For A,B(?)Z, let A+B={α+b:α∈A,b,∈B}.If A+B=Z, the set A is called an additive complement to B in Z. If A is an additive complement to B and no proper subset of A is an additive complement to B, the set A is called a minimal additive complement to B.In2011, Nathanson posed the following problem:Let W be an infinite set of integers. Does there exist a minimal additive complement to W in Z? Does there exist an additive complement to W in Z that does not contain a minimal additive complement?In this thesis, we prove the following results (Published in SIAM J. Discrete Math.):Theorem3. Let W be a set of integers with inf W=-∞and sup W=+∞. Then there exists a minimal additive complement to W.By the next theorem, we can see that, if inf W>-∞or sup W<+∞oo (in this case, take W1=-W, then inf W1>-∞),then the conclusion above is not true. Theorem4.Let W={l=ω1<ω2<…}be a set of integers and W=(z∩(0,+∞))\W={ω1<ω2<...).(a)If lim sup(ωi+1-ωi)＝+∞,then there exists a minimal additive comple-ment to W;i�'+∞(b)If lim i�'+∞(ωi+1-ωi)=+∞,then there does not exist a minimal additive complement to W.