On an additive property of sequences of nonnegative integers

Periodica Mathematica Hungarica - Let a1&;lt;... be an infinite sequence of positive integers, let k≥2 be a fixed integer and denote by Rk(n) the number of solutions of n=ai1+ai2+...+aik....     

On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

On small sets of integers

The Ramanujan Journal - An upper quasi-density on $$\mathbf{{H}}$$ (the integers or the non-negative integers) is a real-valued subadditive function $$\mu ^\star $$ defined on the whole power set...     

On a combinatorial method for counting smooth numbers in sets of integers

In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums a+b, a∈A, b∈B. The proof of this result is completely combinatorial and elementary.     

A short proof of a theorem on degree sets of graphs

For a finite simple graph G, we denote the set of degrees of its vertices, known as its degree set, by D(G). Kapoor, Polimeni and Wall [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] have determined the least number of vertices among graphs with a given degree set. We give a very short proof of this result.     

On primitive sets of squarefree integers

Let PN(resp. P*N) be the family of the primitive subsets of f{1, 2, ... N } (resp. the squarefree integers not exceeding N). We prove the following conjecture (even in a more general form) of Pomerance and Sárközy ... In a new direction we obtain surprisingly sharp estimates for ... As a common generalization we present conjectures about ...     

On certain other sets of integers

We show that if A ⊂ {1,...,N} contains no non-trivial three-term arithmetic progressions then |A| = O(N/log^3/4−o(1) N).     

On the closure of sets of attainability in ℝ2

We display a class of control systems in the plane for which it is generically true that the set attainable from any point in the plane is a closed Whitney prestratified set. That is, even if the attainable set of some given system is not closed, we can approximate the system as closely as we wish by another system whose attainable sets are closed.     

On sets of integers with prescribed gaps

For a fixed setI of positive integers we consider the set of paths (p _o,...,p _k ) of arbitrary length satisfyingp _l –p _l–1I forl=2,...,k andp ₀=1,p _k =n. Equipping it with the uniform distribution, the random path lengthT _n is studied. Asymptotic expansions of the moments ofT _n are derived and its asymptotic normality is proved. The step lengthsp _l –p _l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp ₀ andp _k are not specified. In the special caseI={m+1,m+2,...} (for some fixed m) we derive the exact distribution of a random m-gap subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented.     

On multiple sum and product sets of finite sets of integers

Let $\text{A?}\text{Z}$ be a finite set of integers of cardinality |A|=N?2. Given a positive integer k, denote kA (resp. A^(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A^(k)|)>N^b. This answers affirmably questions raised in Erd?s and Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the additive completion of sets of integers

Denote by k = k(N) the least integer for which there exists integers b₁, b₂, …, b_k satisfying 0 ≤ b₁ ≤ b₂ ≤ … ≤ b_k ≤ N such that every integer in |1, N| can be written in the form i² + b_j. It is shown that for all sufficiently large N, k ≥ (1.147)√N.     

On some sets of pairs of positive integers

We define a density for sets of pairs [m, n] of positive integers and determine it for some sets defined by arithmetic properties of m and n.We derive results on the irreducible fractions with denominator ≤ x which belong to a fixed given interval.     

On sets of integers with the Schur property

We investigate sets of integers for which Rado and Schur theorems are true from the point of view of their local density. We establish the existence of locally sparse Rado and Schur sets in a strong sense.     

On Steinhaus sets

We give a common proof of several results on Steinhaus sets in for d2 including the fact that a Steinhaus set in must be disconnected.     

On a Class of Uniformly Distributed Sequences of Point Sets

In this note we construct certain sequences of finite point sets in [0, 1)^s(s ≥ 1) and give the upper bounds of their discrepancy. Furthermore we prove that these sequences are uniformly distrbuted in [0, 1)^s.