Sum Of Digits,Ordinary Integers And Problems Of Sumsets

In this thesis we investigate the sum of digits of polynomial values, integers with a given number of divisors and arithmetic progressions in sumsets and difference sets. The main results are as follows:1. The sum of digits of polynomial valuesFor any integer q≥ 2, denote by sq(n) the sum of the digits in the q-ary expan-sion of a nonnegative integer n.Many researchers have done deep research on this topic. In this thesis we inves-tigate the ratios of the sums of digits of polynomial values.Assume that with h≥ 1,l≥ 1, ah> 0 and bl> 0. We prove the following theorems:Theorem 1.1. Let deg p1(x)> deg p2(x). If p1(n)≥ 1 and p2(n)≥ 1 for any positive integer n, then for any ε> 0, there exists a positive constant C1, depending only on ε, q, p1(x) and p2(x), such that for all sufficiently large integers N, where α= ε{2h{l+3)(h[l+3)+1)+ε)-1.Theorem 1.2. Let deg pi(x)< deg p2(x). If p1(n)≥ 1 and p2(n)≥ 1 for any postive integer n, then for any ε> 0, there exists a positive constant C2, depending only on ε, q, p1(x) and p2(x), such that for all sufficiently large integers N.2. Integers with a given number of divisorsFor any positive integer n> 1, let n= q1q2... qs be the prime factorization of n with q14≥ q2≥…≥ qs> 1. A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p1q1-1p2q2-1...psqs-1, where pk denotes the κth prime for any integer k with 1≤κ≤s. A positive integer n is said to be extraordinary if n>1 and n is not ordinary.In 1968, Grost [26] determined all extraordinary integers n with Ω(n)≤ 6 and proved that 16p is an extraordinary integer for any odd prime p.In 2006, Brown [5] proved the following results:(i) A prime power pk is extraor-dinary if and only if 2P≤pk, where pκ denotes the κth prime; (ii) All square-free positive integers are ordinary; (iii) Let O be the set of all ordinary integers, then for any real number δ with 0<δ<1/2, we have Specially, the set of all ordinary integers has asymptotic density one.In this thesis, we find several sufficient conditions for positive integers being ordinary integers, improve the upper bound of the counting function of extraordinary integers, and determine all the ordinary integers of 5-free integers. We prove the following three theorems, which are published in J. Number Theory.Theorem 2.1. Let n=q1q2…qs be the prime factorization of n with q1≥q2≥…≥qs>1. If for all integers i with 1≤i≤(?), where pk denotes the kth prime, and [(?)] denote by the least integer not less than (?), then n is ordinary.Theorem 2.2.For any ε>0,we have N exp(-log log N)≤|[1,N]\O|≤N exp(-(loglogN)1-ε) for all sufficiently large integers N,where[1,N]={1,2,…,N).Theorem 2.3.A 5-free integer is extraordinary if and only if it is in F5,whereF5 ={2×34,22×33,22×34,22×34×5,23,23×3,23×32,23×33, 23×34,23×34×5.24,24×32,24×33,24×34,24×33×5, 24×34×5,24×34×54}∪{16p:p is an odd prime}.3.Arithmetic progressions in sumsets and difference setsFor A, B(?)[1,N]={1,2,…,N},let A+B={a+b:a∈A,b∈B},and A-B={a-b:a∈A,b∈B).In 2007,Croot,Ruzsa and Schoen[9]proved that for every odd integer κ≥3 and sufficiently large integer N,if A,B(?)[1,N] with |A||B|≥6N2-2/(κ-1),then A+B contains an arithmetic progression of length κ.In 2010,Hamel,Lyall,Thompson and Walters[28]proved that for every odd integer κ≥3 and sufficiently large integer N,if A(?)[1,N] with |A|≥4N1-2/(κ-1), then A-A contains an arithmetic progression of length κ.In this thesis,we prove the following two theorems,which are published in Int. J.Number Theory.Theorem 3.1.Let s ≥ 1,Ai(?)[1,N](1≤i ≤ s) and κ≥3 be an odd integer.If |A1||A2|…|As|≥2s-1(?)Ns-2/(κ-1),then(a)each of Ai-Aj(1≤i,j≤s)contains an arithmetic progression of length κ with the same common difference;(b)all sets Ai-Ai(1≤i≤s) contain a common arithmetic progression of length κ.Theorem 3.2.For any δ>0 and any odd number k≥3,there exists an explicit constant N0=N(δ,k)such that if N≥N0 and A,B(?)[1,N]with[A||B|≥ δN2-2/(k-1).then A+B contains an arithmetic progression of length k.