| In this thesis we investigate the additive decomposition of the set of all quadratic residues modulo a prime,ε-harmonic numbers and additive representation functions.1.A conjecture of Sarkozy on additive decompositionsFor any prime p,let Rp be the set of all quadratic residues modulo p.If there exist A1,…,Ak with |A1|,…,|Ak|…≥ 2 such that Rp=A1+…+Ak,then we call it an additive k-decomposition of Rp.In 2012,Sarkozy[40]conjectured that,for all sufficiently large primes p,Rp has no additive 2-decomposition.He also proved that if Rp=U+V is an additive 2-decomposition for a sufficiently large prime p,then#12 In 2004,Shkredov[45]proved that#12In the first chapter,we further improve this result for any prime p.(1)For any prime p,if Rp=U+V is an additive 2-decomposition of Rp,then(?)(2)For any prime p>5381,Rp has no additive 3-decomposition.(3)For any prime p,Rp has no additive 4-decomposition.2.ε-harmonic numbers For any positive integer n and any sequence of integers ε={εi}i=1∞,let#12 The number Hn,ε is called the n-th ε-harmonic number.In 2019,Wu and Chen[53]posed following problem:for any sequence ε=ε={εi}i=1∞ with εi∈{1,-1}(i=1,2,…),does the set of positive integers n with bn,ε=bn+1,ε-have density one?In the second chapter,we prove that if ε={εi}i=1∞ is a pure recurring sequence with εi∈{1,-1}(i=1,2,…),then the set of positive integers n with bn,ε=bn+1,εhas density one.Let Hn be the n-th ε-harmonic number with εi=1(i=1,2,…),un be the numerator of Hn and let vn be the denominator of Hn.Clearly,vp(Hn)≥-vp(vn)≥-[logp n].Let Tp(k)be the set of all positive integers n with vp(Hn)=-[logp n]+k.In 2007,Wu and Chen[51]provided the formula of the logarithmic density of Tp(0).In the second chapter,we provide the formula of the logarithmic density of Tp(k)for all integers k≥1.Let Jp be the set of all positive integers n with p |un.Eswarathasan and Levine[18]conjectured that Jp is finite for every prime p.If the conjecture is true,by the formula of the logarithmic density of Tp(k),then Tp(k)has logarithmic density zero for all sufficiently large integers k.3.Additive representation functionsFor any nonnegative integer set A and any nonnegative integer n,let R1(A,n),R2(An),R3(A,n)be defined as the number of solutions of the equation n=a+a’,a,a’E A;n=a+a’,a,a’∈A,a<a’;n=a+a’,a,a’∈A,a≤a’,respectively.For i ∈ {1,2,3},Sarkozy ever asked if there exist nonnegative integer sets A,B such that|(A ∪ B)\(A ∩ B)|=∞ and Ri(A,n)=Ri(B,n)for all sufficiently large integers n.We name this problem as Sarkozy’s problem.Because the answer of Sarkozy’s problem for i=1 is negative,we obtain the necessary and sufficient condition for the structure of the sets A,B with A ∪ B=N and A∩B=(?)such that |R1(A,n)-R1(B,n)|≤1 holds for all nonnegative integers n in the third chapter.This result has been published in Bull.Aust.Math.Soc..In 2008,Tang[48]asked if we can divide N into the union of k(≥3)disjoint sets A1,…,Ak such that the equation R3(A1,n)=…=R3(Ak,n)holds for all sufficiently large integers n.In the third chapter,we prove that the equation cannot hold for some integers n≥2k.This result has been published in Int.J.Number Theory.In addition,we improve the result of Kiss and Sandor[31]about representation functions and it is the best possible up to constant factors.This result has been pub-lished in Publ.Math.Debrecen. |
