On the length of arithmetic progressions in linear combinations of S-units

Recent finiteness results concerning the lengths of arithmetic progressions in linear combinations of elements from finitely generated multiplicative groups have found applications to a variety of problems in number theory. In the present paper, we significantly refine the existing arguments and give an explicit upper bound on the length of such progressions.     

The closures of arithmetic progressions in the common division topology on the set of positive integers

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.     

On the Simplest Inverse Problem for Sums of Sets in Several Dimensions

d -dimensional sets having the smallest cardinality of the sum set. Let be a finite d-dimensional set such that . If , then K consists of d parallel arithmetic progressions with the same common difference. We also establish the structure of K in the remaining cases . Received: February 5, 1996/Revised: November 20, 1997     

Powerful arithmetic progressions

We give a complete characterization of so-called powerful arithmetic progressions, i.e. of progressions whose kth term is a kth power for all k. We also prove that the length of any primitive arithmetic progression of powers can be bounded both by any term of the progression different from 0 and ±1, and by its common difference. In particular, such a progression can have only finite length.     

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′).     

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.     

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.     

N-graphs,Modular Sidon and Sum-Free Sets,and Partition Identities

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.     

Generalized van der Waerden numbers

Certain generalizations of arithmetic progressions are used to define numbers analogous to the van der Waerden numbers. Several exact values of the new numbers are given, and upper bounds for these numbers are obtained. In addition, a comparison is made between the number of different arithmetic progressions and the number of different generalized arithmetic progressions.     

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.     

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.     

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....     

Representing algebraic integers as linear combinations of units

In this paper we consider representations of algebraic integers of a number field as linear combinations of units with coefficients coming from a fixed small set, and as sums of elements having small norms in absolute value. These theorems can be viewed as results concerning a generalization of the so-called unit sum number problem, as well. Beside these, extending previous related results we give an upper bound for the length of arithmetic progressions of \(t\) -term sums of algebraic integers having small norms in absolute value.     

On arithmetic progressions in recurrences – A new characterization of the Fibonacci sequence

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given.     

The structure of popular difference sets

We show that the set of popular differences of a large subset of ℤ _N does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa in [Ruz87] and later used in [Ruz91] to show that there exists a large subset of ℤ _N whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen’s Inequality.     

Arithmetic on Groups of Positive Rationals

In this paper we develop a theory of unique factorization for subgroups of the positive rationals. We show that this theory is strong enough to include arithmetic progressions and the theory of genera in algebraic number fields. We establish generalizations of both Dirichlet's theorem on primes in arithmetic progressions and the theory of genera for Abelian extensions of the rationals.     

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.     

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.     

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.