Connections between connected topological spaces on the set of positive integers

In this paper we introduce a connected topology T on the set ? of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ? which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (?, T) and (?, T′).     

On arithmetic progressions on Pellian equations

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x ² ? dy ² = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.     

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.     

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.     

Arithmetic progressions on Pell equations

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X,Y) whose X-coordinates or Y-coordinates are in arithmetic progression.     

Tessellating polyominos in the plane

Let the R² space be divided into unit squares where a polyomino is a finite, connected set of unit squares. In this paper, we give a necessary and sufficient condition on tessellating polyominos by observing an unexpected relation between such tessellations and systems of arithmetic progressions.     

On the distribution of square-full numbers in arithmetic progressions

A positive integer n is called a square-full number if p ² divides n whenever p is a prime divisor of n. In this paper we study the distribution of square-full numbers in arithmetic progressions by using the properties of Riemann zeta functions and Dirichlet L-functions.     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

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.     

B-free numbers in short arithmetic progressions

Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions.     

The complement of certain recursively defined sets

An RD-set (recursively defined) is a minimal set of positive integers containing a given seed and closed under a given set of unary linear operations (x → ax + b). We examine conditions under which the complement of an RD-set may be expressed as the disjoint union of infinite arithmetic progressions.     

Infinite families of infinite families of congruences for k-regular partitions

Let k∈{10,15,20}, and let b _k (n) denote the number k-regular partitions of n. We prove for half of all primes p and any t≥1 that there exist p?1 arithmetic progressions modulo p ^2t such that b _k (n) is a multiple of 5 for each n in one of these progressions.     

Some new results on k-free numbers

In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban-Davenport-Halberstam type for the k-free numbers in arithmetic progressions.     

An uncertainty principle for function fields

In a recent paper, Granville and Soundararajan (2007) [5] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be well-distributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of Granville and Soundararajan (2007) [5] and prove that a similar phenomenon holds in Fq[t].     

An upper bound for van der Waerden-like numbers usingk colors

Functions analogous to the van der Waerden numbers w(n, k) are considered. We replace the class of arithmetic progressions,A, by a classA′, withA ? A′; thus, the associated van der Waerden-like number will be smaller forsi’. We consider increasing sequences of positive integers x₁,…, x_n which are either arithmetic progressions or for which there exists a polynomial φ(x) with integer coefficients satisfying φ(xi) = x_i+1, i = 1,…,n - 1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w′(n, k) for the general pair (n,k). A table of several new computer-generated values of these functions is provided.     

Convergent sieve sequences in arithmetic progressions

Let V be a set of pairwise coprime integers not containing 1 and suppose, there is a 0?δ<1, such that _∑v∈Vv−1+δ<∞ holds. Let χV(n)=1 if v?n for all v∈V and χV(n)=0 elsewhere. We study the behavior of χV in arithmetic progressions uniformly in the modulus, both individually and in the quadratic mean over the residue classes. As an application, new bounds for the mean square error of squarefree numbers in arithmetic progressions are obtained.     

Long arithmetic progressions in critical sets

Given a density 0<σ?1, we show for all sufficiently large primes p that if S⊆Z/pZ has the least number of three-term arithmetic progressions among all sets with at least σp elements, then S contains an arithmetic progression of length at least log^1/4+o(1)p.     

Arithmetic Progressions and Its Applications to (m,q)-Isometries: A Survey

In this paper we collect some results about arithmetic progressions of higher order, also called polynomial sequences. Those results are applied to (m, q)-isometric maps.     

Maximal subsets free of arithmetic progressions in arbitrary sets

The problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length k in a given set of size n is considered. It is proved that it is sufficient, in a certain sense, to consider the interval [1,..., n]. The study continues the work of Komlós, Sulyok, and Szemerédi.     

The distribution of quadratic residues and nonresidues in arithmetic progressions

We prove some results concerning the distribution of quadratic residues and nonresidues in arithmetic progressions in the setting \( {{\mathbb{F}}_p}={{\mathbb{Z}} \left/ {{p\mathbb{Z}}} \right.} \) , where p is a large prime.