Sub-Ramsey Numbers for Arithmetic Progressions

Let the integers 1, . . . ,n be assigned colors. Szemerédi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing totally multicolored arithmetic progressions of length 3. Let f(n) be the smallest integer k such that there is a coloring of {1, . . . ,n} without totally multicolored arithmetic progressions of length three and such that each color appears on at most k integers. We provide an exact value for f(n) when n is sufficiently large, and all extremal colorings. In particular, we show that f(n)=8n/17+O(1). This completely answers a question of Alon, Caro and Tuza.     

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.     

Maximal Collections of Intersecting Arithmetic Progressions

Let N _t (k) be the maximum number of k-term arithmetic progressions of real numbers, any two of which have t points in common. We determine N ₂(k) for prime k and all large k, and give upper and lower bounds for N _t (k) when t 3.* Research supported in part by NSF grant DMS-0070618.     

Discrepancy One among Homogeneous Arithmetic Progressions

We investigate a restriction of Paul Erd?s’ well-known problem from 1936 on the discrepancy of homogeneous arithmetic progressions. We restrict our attention to a finite set S of homogeneous arithmetic progressions, and ask when the discrepancy with respect to this set is exactly 1. We answer this question when S has size four or less, and prove that the problem for general S is NP-hard, even for discrepancy 1.     

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     

算术级数中的奇数Goldbach问题

本文给出了算术级数的模的精确数值上界,在该算术级数中奇数Goldbach问题可解。我们的结果蕴含了Linnik常数的一个数值上界。     

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.     

Discrepancy of Arithmetic Progressions in Higher Dimensions

K. F. Roth (1964, Acta. Arith.9, 257-260) proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, …, N} is at least cN^1/4, and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N]^d and prove that this result is nearly sharp. We use our results to give an upper estimate for the discrepancy of lines on an N×N lattice, and we also give an estimate for the discrepancy of a related random hypergraph.     

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…     

关于三次素变数方程在算术数列中的解

In this paper, we give a necessary and sufficient solvable condition for diagonal cubic equation with prime variable in arithmetic progressions and the outline of the proof.     

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.     

A Variance for k-Free Numbers in Arithmetic Progressions

The second moment for the the distribution of k-free numbersinto arithmetic progression is studied and the asymptotic formulaof Hooley-Montgomery type is obtained for a wider range of valuesfor the modulus than hitherto. In the special case of squarefreenumbers this improves on the theorem of Warlimont from 1980.2000 Mathematics Subject Classification 11N25.     

算术级数中的华罗庚五素数平方定理

本文给出了华罗庚五素数平方定理的算术级数形式,证明了其中一个素数可 以取在大模的算术级数中.     

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.     

On the Existence of Rainbow 4-Term Arithmetic Progressions

For infinitely many natural numbers n, we construct 4-colorings of [n]  =  {1, 2, ..., n}, with equinumerous color classes, that contain no 4-term arithmetic progression whose elements are colored in distinct colors. This result solves an open problem of Jungić et al. (Comb Probab Comput 12:599–620, 2003) Axenovich and Fon-der-Flaass (Electron J Comb 11:R1, 2004).     

算术级数中的五素数平方和定理

设N是充分大的正整数满足N≡5mod 24,l和d是满足(l,d)=1的整数.A0,A>1是满足A0=600A+2000的正常数.本文证明对所有的整数0相似文献   

Parity of the Partition Function in Arithmetic Progressions, II

Let p(n) denote the ordinary partition function. Subbarao conjecturedthat in every arithmetic progression r (mod t) there are infinitelymany integers N = r (mod t) for which p(N) is even, and infinitelymany integers M = r (mod t) for which p(M) is odd. We provethe conjecture for every arithmetic progression whose modulusis a power of 2. 2000 Mathematics Subject Classification 11P83.