Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We study in this paper a new duality identity between large and small prime factors of integers and its relationship with the prime number theorem for arithmetic progressions. The asymptotic behavior of large prime factors of integers leads to interesting relations involving the Möbius function.     

Concordant sequences and concordant entire functions

A theorem of Pólya shows that the function 2 ^Z is the ‘smallest’ transcendental entire function that is integer valued on the set ℕ of non-negative integers. Analogous results have been established in which ℕ is replaced by other sets of integers, beginning with the result of Gel’fond for geometric sequences of integers. Other results consider the imposition of additional congruence conditions on the value sequence of the candidate entire function on the subject sequence. The present paper extends the consideration of such congruence conditions from ℕ and geometric sequences to more general sets for which Pólya-type results have been established.     

Sumsets contained in infinite sets of integers

If the positive integers are partitioned into a finite number of cells, then Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition. Erdös conjectured that if A is a set of integers with positive asymptotic density, then there exist infinite sets B and C such that B + C ? A. This conjecture is still unproved. This paper contains several results on sumsets contained in finite sets of integers. For example, if A is a set of integers of positive upper density, then for any n there exist sets B and F such that B has positive upper density, F has cardinality n, and B + F ? A.     

Integers with consecutive divisors in small ratio

An elementary construction of a sequence of positive integers is given. The sequence settles a question of Erdös concerning integers with consecutive divisors in small ratio.     

n-Tuples of positive integers with the same second elementary symmetric function value and the same product

In this paper, by using the theory of elliptic curves, we prove that for every k, there exist infinitely many primitive sets of k n-tuples of positive integers with the same second elementary symmetric function value and the same product.     

Representing powers of 2 by a sum of four integers

We solve an arithmetic problem due to Erdös and Freud (1986) investigated also by Freiman, Nathanson and Sárközy: How many elements from a given set of integers one must take to represent a power of 2 by their sum?     

Additive Partitions and Continued Fractions

A set S of positive integers is avoidable if there exists a partition of the positive integers into two disjoint sets such that no two distinct integers from the same set sum to an element of S. Much previous work has focused on proving the avoidability of very special sets of integers. We vastly broaden the class of avoidable sets by establishing a previously unnoticed connection with the elementary theory of continued fractions.     

Generalized Sidon sets

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.     

The number of certain integral polynomials and nonrecursive sets of integers, Part 2

We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, from recursion theory, and from the negative solution to Hilbert's 10th Problem.

     

On the Number of Prime Facttors of Integers Characterized by Digit Properties

The number of distinct prime factors of integers with missing digits is considered, and both the normal order and large values of the function over sets of this type are studied. A conjecture of Mauduit and Sárközy, on large values of the function over integers whose sum of digits is fixed, is also proved.     

A Generalization of the Siegel–Walfisz Theorem and its Application

We investigate the distribution of integers with a fixed number of prime factors in arithmetic progressions, and obtain a generalization of the Siegel–Walfisz theorem under the extended Riemann hypothesis. As an application, we consider a problem of P. Erd?s, A. M. Odlyzko and A. Sárközy about the representation of residue classes modulo q by products of two integers with a fixed number of prime factors. We show some conditional results.     

Singularity sets of Lévy processes

We completely describe the size and large intersection properties of the Hölder singularity sets of Lévy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Lévy process. The Hölder singularity sets of the sample paths of certain random wavelet series are investigated as well.     

On the Exponential sum over r-Free Integers

We determine, up to a constant factor, the L ¹ mean of the exponential sum formed with the r-free integers. This improves earlier results of Brüdern, Granville, Perelli, Vaughan and Wooley. As an application, we improve the known bound for the L ¹ norm of the exponential sum defined with the Möbius function.     

Allowable graphs of the nonlinear Schrödinger equation and their applications

We construct the definition of allowable graphs of the nonlinear Schrödinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving 3-dimensional blocks of the normal form of the nonlinear Schrödinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schrödinger equation.     

Remarks on Steinhaus' property and ratio sets of sets of positive integers

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.     

Countably infinite quasi-convex sets in some locally compact abelian groups

We produce a class of countably infinite quasi-convex sets (sequences converging to zero) in the circle group T and in the group J₂ of 2-adic integers determined by sequences of integers satisfying a mild lacunarity condition. We also extend our results to the group R of real numbers. All these quasi-convex sets have a stronger property: Every infinite (necessarily) symmetric subset containing 0 is still quasi-convex.     

Consecutive integers in high-multiplicity sumsets

Sharpening (a particular case of) a result of Szemerédi and Vu [4] and extending earlier results of Sárközy [3] and ourselves [2], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers, whose length is comparable with the lengths of the set summands.A corollary of our main result is as follows. Let k,l≥1 and n≥3 be integers, and suppose that A ₁,…,A _k ?[0,l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k≥2?(l?1)/(n?2)?, then the sumset A ₁+???+A _k contains a block of at least k(n?1)+1 consecutive integers.     

Additive and multiplicative Sidon sets

Summary Inspired by a paper of Sárk?zy [4] we study sets of integers and sets of residues with the property that all sums and all products are distinct.     

Erdös Conjecture on Sums of Distinct Numbers

A conjecture of Erdös that a set of n distinct numbers having the most linear combinations with coefficients 0,1 all equal are n integers of smallest magnitude is here proven. The result follows from a theorem of Stanley that implies that the integers from n have the most such linear combinations having k distinct values for every k. The same result is shown to hold for complex numbers and vectors in Hilbert space. It is shown that the number of linear combinations taking on k distinct values is maximized by the same configuration, for every k. Generalization to the case in which irregular distinctness restrictions are imposed is also given.     

Finiteness results for F-Diophantine sets

Monatshefte für Mathematik - Diophantine sets, i.e. sets of positive integers A with the property that the product of any two distinct elements of A increased by 1 is a perfect square, have a...