Arithmetic Progressions in Sets of Fractional Dimension

Let E ì \mathbbR{E \subset\mathbb{R}} be a closed set of Hausdorff dimension α. Weprove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.     

An Additive Problem with Primes in Arithmetic Progressions

In this paper, we extend a classical result of Hua to arithmetic progressions with large moduli. The result implies the Linnik Theorem on the least prime in an arithmetic progression.     

Spherical Designs and Generalized Sum-Free Sets in Abelian Groups

We extend the concepts of sum-freesets and Sidon-sets of combinatorial number theory with the aimto provide explicit constructions for spherical designs. We calla subset S of the (additive) abelian group G t-free if for all non-negative integers kand l with k+l t, the sum of k(not necessarily distinct) elements of S does notequal the sum of l (not necessarily distinct) elementsof S unless k=l and the two sums containthe same terms. Here we shall give asymptotic bounds for thesize of a largest t-free set in Z _n,and for t 3 discuss how t-freesets in Z _n can be used to constructspherical t-designs.     

The Ternary Goldbach Problem with Primes in Arithmetic Progressions

In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik constant. Received November 4, 1998, Accepted March 6, 2001     

New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.     

Probabilistic Number Theory in Additive Arithmetic Semigroups, III

We extend the investigation of quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups given in our previous paper. Then the new and old quantitative mean-value theorems are applied to the investigation of local distribution of values of a special additive function *(a). The result is unexpected from the point of view of classical number theory. This reveals the fact that the essential divergence of the theory of additive arithmetic semigroups from classical number theory is not related to the existence of a zero of the zeta function Z(y) at y = –q ^–1.     

On the Distribution of Values of Euler's Function over Integers in Arithmetic Progressions

Letφ(n) denote the Euler-totient function, we study the distribution of solutions ofφ(n)≤x in arithmetic progressions, where n≡l(mod q) and an asymptotic formula was obtained by Perron formula.     

On Assouad Dimension and Arithmetic Progressions in Sets Defined by Digit Restrictions

Journal of Fourier Analysis and Applications - We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one....     

Arithmetic Progressions,Prime Numbers,and Squarefree Integers

In this paper we establish the distribution of prime numbers in a given arithmetic progression p l for which ap + b is squarefree.     

Mean-Value Theorems in Arithmetic Semigroups

This paper reports on recent progress in the theory of multiplicative arithmetic semigroups, which has been initiated by John Knopfmacher's work on abstract analytic number theory. In particular, it deals with abstract versions of the mean-value theorems of Delange, of Wirsing, and of Halász for multiplicative functions on arithmetic semigroups G with Axiom A . The Turán Kubilius inequality is transferred to G , and methods developed by Rényi, Daboussi and Indlekofer, Lucht and Reifenrath are utilized. As byproduct a new proof of the abstract prime number theorem is obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Arithmetic progressions in self-similar sets

Given a sequence {\{b_i\}}_{i=\mathrm{1}}^n and a ratio \lambda \in (\mathrm{0},\mathrm{1}), let E={\cup }_{i=\mathrm{1}}^n(\lambda E+b_i) be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in E: Our main idea is from the multiple β-expansions.     

Herbrandizing search problems in Bounded Arithmetic

We study search problems and reducibilities between them with known or potential relevance to bounded arithmetic theories. Our primary objective is to understand the sets of low complexity consequences (esp. Σ^b₁ or Σ^b₂) of theories Sⁱ₂ and Tⁱ₂ for a small i, ideally in a rather strong sense of characterization; or, at least, in the standard sense of axiomatization. We also strive for maximum combinatorial simplicity of the characterizations and axiomatizations, eventually sufficient to prove conjectured separation results. To this end two techniques based on the Herbrand's theorem are developed. They characterize/axiomatize Σ^b₁‐consequences of Σ^b₂‐definable search problems, while the method based on the more involved concept of characterization is easier and gives more transparent results. This method yields new proofs of Buss' witnessing theorem and of the relation between PLS and Σ^b₁(T¹₂), and also an axiomatization of Σ^b₁(T²₂). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the density of integers of the form 2 k + p in arithmetic progressions

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k ＋ p, and give a quantitative form of Romanoff＇s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative.     

On the Least Prime in Some Arithmetic Progressions

Supposethatqisasufficientlylargepositiveinteger,(a,q)=1,letP(a,q)betheleastprimeinthearithmeticprogression{n=a(modq)}andxbetheDirichletcharacterofmodulusq.L(s,x)istheDirichletL-function.Everypositiveintegerqcanbeexpressedasq=qq2,q2iscubefree.Anintegerqiscalled"hasboundedcubicpart"iftheaboveq3isboundedabsolutely.Forgeneralintegerq,Heath-Brown[1]showedthatP(a,q)q5.5.Whenqisaprime,Motohashi[2]showedthat:Foranyfordpositiveintegeraandsmallpositiverealnumber>0,thereexistindnitelymanyprimesqsucht…     

等差项乘积的渐进估计

研究了等差项乘积∏ni=1a_i的渐进估计.首先给出了一系列关于等差项乘积的不等式,继而应用Euler-Maclaurin求和公式及Γ函数的Stirling公式:Γ(x+1)~(2πx)~(1/2)(x/e)~x(x→+∞),推导出了∏ni=1a_i的较精确的渐进式,最后,得到了精确化的Wallis公式.     

Arithmetic Progressions in Salem-Type Subsets of the Integers

Given a subset of the integers of zero density, we define the weaker notion of the fractional density of such a set. We show that a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density whose characteristic functions have Fourier coefficients that decay sufficiently rapidly.     

Small Sets which meet all the n-Term Arithmetic Progressions in the Interval [1, n2]

For any positive integers n and k, let f(n, k) denote the smallestsize of a subset of the integer interval I =[l, n] which meetsall the k-term arithmetic progressions contained in I. We showthat n+(1/2)n^1/2–2 < f(n²,n) , where p is the largest prime n, and for any real number x,[x] is the least integer x.     

不可约特征标次数为等差数的有限可解群

设G为有限群,cd(G)表示G的所有复不可约特征标次数的集合.本文研究了不可约特征标次数为等差数的有限可解群,得到两个结果:如果cd(G)={1,1+d,1+2d,…,1+kd},则k≤2或cd(G)={1,2,3,4};如果cd(G)={1,a,a+d,a+2d,…,a+kd},|cd(G)|≥4,(a,d)=1,则cd(G)={1,2,2^e+1,2e+1,2^(e+1)},并给出了d>1时群的结构.