Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

In this paper we use the Nash-Williams theory of fronts and barriers to study weakly null sequences in Banach spaces. Specifically, we show how barriers relate to the classical fact that C(K) with K a countable compactum is c₀-saturated. Another result relates the notion of a barrier to the Maurey-Rosenthal example of a weakly null sequence with no unconditional subsequences. In particular, we construct examples of weakly-null sequences which are α-unconditional but not β-unconditional.     

On smooth sets of integers

This work studies evenly distributed sets of integers—sets whose quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation. Namely, for ρ,Δ∈R a set A of integers is (ρ,Δ)- smooth if for any interval I of integers; a set A is Δ-smooth if it is (ρ,Δ)-smooth for some real number ρ. The paper introduces the concept of Δ-smooth sets and studies their mathematical structure. It focuses on tools for constructing smooth sets having certain desirable properties and, in particular, on mathematical operations on these sets. Three additional papers by us are build on the work of this paper and present practical applications of smooth sets to common and well-studied scheduling problems.One of the above mathematical operations is composition of sets of natural numbers. For two infinite sets A,B⊆N, the composition of A and B is the subset D of A such that, for all i, the ith member of A is in D if and only if the ith member of N is in B. This operator enables the partition of a (ρ,Δ)-smooth set into two sets that are (ρ₁,Δ)-smooth and (ρ₂,Δ)-smooth, for any ρ₁,ρ₂ and Δ obeying some reasonable restrictions. Another powerful tool for constructing smooth sets is a one-to-one partial function f from the unit interval into the natural numbers having the property that any real interval X⊆[0,1) has a subinterval Y which is ‘very close’ to X s.t. f(Y) is (ρ,Δ)-smooth, where ρ is the length of Y and Δ is a small constant.     

Bases for sets of integers

We are interested in expressing each of a given set of non-negative integers as the sum of two members of a second set, the second set to be chosen as economically as possible.So let us call B a basis for A if to every a ∈ A there exist b, b′ ∈ B such that a = b + b′. We concern ourselves primarily with finite sets, A, since the results for infinite sets generally follow from these by the familiar process of condensation.     

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...     

Estimates related to sumfree subsets of sets of integers

A subsetA of the positive integers ?₊ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?₊ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines _k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) _k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys _k (B)>δ _k |B|, valid for any finite subsetB of ?₊, necessarilyδ _k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.     

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 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 ...     

Linear equations and sets of integers

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then |A|\leqq 2^-ck/(logk)2N|A|\leqq 2^{-ck/{(\log k)}^{2}}N, for some absolute constant c>0, provided that N is large enough.     

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 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.     

Powers of cyclotomic integers and difference sets

Let p be an odd prime, $\text{ζ =}\text{rxp}\text{(}\text{2πi}\text{p}\text{)}$, D a difference set mod p having a nontrivial multiplier, and ν = H(ζ), where H(x) is the Hall polynomial of D. For any α = Σ_i=0^p?1a_iζⁱ with rational a_i denote δ(α) = max ∥ a_i ? a_j ∥. Assuming that there are no nontrivial multiplicative dependence relations among the conjugates of ν, we obtain results for

. We then show that for most known families of difference sets mod p the required independence result is valid. A conjecture concerning the exact value of the first number is stated. The conjecture is confirmed in certain particular cases.     

On some random thin sets of integers

We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d'Analyse Mathématique 86 (2002), 105-138, namely that there exist -Rider sets which are sets of uniform convergence and -sets for all but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for , the -Rider sets which we had constructed in that paper are almost surely not of uniform convergence.

     

Multiplicative properties of sets of positive integers

It is proved that every set of positive integers with upper Dirichlet density greater than 1/2 contains three distinct elements whose product is a square. Several similar problems are considered.     

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 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.