A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

Let λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ _∞ ^λ -spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = ℓ₂, we obtain a ℒ _∞ ^λ -spaceX with the RNP such that the projective tensor product containsc ₀ and hence fails the RNP. TakingE=L ¹, we obtain a ℒ _∞ ^λ -space failing the RNP but nevertheless not containingc ₀.     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type under nearly optimal conditions on . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of , obtaining a nontrivial estimate provided . This is a further improvement on [BGK]. Received: May 2007, Revision: October 2007, Accepted: October 2007     

Exponential sum estimates over subgroups and almost subgroups of \mathbb{Z}_{Q}^{*} , where Q is composite with few prime factors

In this paper we extend the exponential sum results from [BK] and [BGK] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q^δ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand, the methods from [BGK] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog–Szemeredi theorem. As a corollary, we get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p^r (p prime) for all r. Only the case r = 2 has been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [KS]. Submitted: October 2004 Revision: June 2005 Accepted: August 2005     

Sieving and expanders

Let V be an orbit in ${\mathbb{Z}}^n$ of a finitely generated subgroup Λ of ${\mathrm{GL}}_n(\mathbb{Z})$ whose Zariski closure $\mathrm{Zcl}(\Lambda )$ is suitably large (e.g. isomorphic to ${\mathrm{SL}}_2$). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the ‘congruence graphs’ that we associate with V. This expansion property is established when $\mathrm{Zcl}(\Lambda )={\mathrm{SL}}_2$. To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A Gauss sum estimate in arbitrary finite fields

We establish bounds on exponential sums ${\sum }_{x\in {\mathbb{F}}_q}\psi (x^n)$ where $q=p^m$, p prime, and ψ an additive character on ${\mathbb{F}}_q$. They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order $(m?2)$. More precisely, a non-trivial estimate is obtained provided n satisfies $\gcd (n,\frac{q?1}{p^{\nu }?1})<p^{?\nu }{q}^{1?\epsilon }$ for all $1?\nu <m$, $\nu |m$, where $\epsilon >0$ is arbitrary. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Sharp thresholds of graph properties, and the -sat problem

Given a monotone graph property , consider , the probability that a random graph with edge probability will have . The function is the key to understanding the threshold behavior of the property . We show that if is small (corresponding to a non-sharp threshold), then there is a list of graphs of bounded size such that can be approximated by the property of having one of the graphs as a subgraph. One striking consequence of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of .

As an application of the main theorem we settle the question of the existence of a sharp threshold for the satisfiability of a random -CNF formula.

An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix some of the conjectures raised in this paper are proven, along with more general results.

     

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.