AR KURLBERG’s scientific contributions

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. This paper contains a brief descrip- tion of the proof that was recently presented elsewhere ((L. Fainsilber, P. Kurlberg, B. Wennberg, preprint 2004)). It also presents the results of numerical experi- ments.

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coecients, then the order of A modulo p is greater than p for all p in a density one subset of the primes.

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N1 for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coecients, then the order of A modulo p is greater than p1 for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N1 for all N in a density one subset of the integers.

We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang (6).