J. Bourgain

activity

20012003

collaborators

4 papers

math.CO2003

On the size of $k$ -fold sum and product sets of integers

Jean Bourgain, Mei-Chu Chang

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$ , such that for every finite set $A$ of integers with cardinality $∣ A ∣ > 1$ , we hav…

math.CO2003

A sum-product estimate in finite fields, and applications

Jean Bourgain, Nets Katz, Terence Tao

Let $A$ be a subset of a finite field $F := Z / q Z$ for some prime $q$ . If $∣ F ∣^{δ} < ∣ A ∣ < ∣ F ∣^{1 - δ}$ for some $δ > 0$ , then we prove the estimate $|A+A| + |A.A| \geq c(δ) |A|^{1+\eps…

math.SP2002

Anderson Localization for Time Quasi Periodic Random Schödinger and Wave Operators

Jean Bourgain, Wei-Min Wang

We prove that at large disorder, with large probability and for a set of Diophantine frequencies of large measure, Anderson localization in $Z^{d}$ is {\it stable} under localiz…

math-ph2001

Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential

J. Bourgain, S. Jitomirskaya

We study regularity properties of the Lyapunov exponent L of quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We pro…

On the size of k-fold sum and product sets of integers

A sum-product estimate in finite fields, and applications

Anderson Localization for Time Quasi Periodic Random Schödinger and Wave Operators

Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential

On the size of $k$ -fold sum and product sets of integers