Previously known algorithms to compute the symmetry group of a cusped hyperbolic three-manifold and to test whether two cusped hyperbolic three-manifolds are isometric do not apply directly to closed manifolds. But by drilling out geodesics from closed manifolds one may compute their symmetry groups and test for isometries using the cusped manifold techniques. To do so, one must know precisely how many geodesics of a given length the closed manifold has. Here we prove the propositions needed to rigorously compute a length spectrum, with multiplicities. We also tabulate the symmetry groups of the smallest known closed hyperbolic three-manifolds.

Euler studied double sums of the form $$ \z(r,s)=\sum_{1\le m<n}{1\over n^sm^r} $$ for positive integers $r$ and $s$, and inferred, for the special cases $r = 1$ or $r+s$ odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of $\z(r, s)$ for real numbers $r$ and $s$. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers $r$ and $s$ with $r+s$ odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.

Let $H_n = 1 + \half + \cdots + {1\over n}$ be the $n$-th partial sum of the harmonic series. A classical result of Wolstenholme states that, if $p > 3$ is prime, the numerator of $H_{p-1}$ is divisible by $p^2$. Here we consider, for a given prime $p$, the set $J_p$ of $n$ for which $p$ divides the numerator of $H_n$. This set $J_p$ had been previously determined for $p = 2,3,5,7$. One of our results is that $J_{11}$ contains exactly $638$ integers, the largest of which is a number of $31$ decimal digits. We determine $J_p$ for all $p < 550$ with three exceptions: $83$, $127$ and $397$.

The computation is based on a new $p$-adically convergent formula for the quantity $H_{pn} - H_n/p$. We describe a probabilistic model for the sets $J_p$, based on branching processes. The model predicts that $|J_p| = O(p^2 (\log\log p)^{2+\eps} )$, and that there are infinitely many $p$ with $|J_p| \ge p^2 (\log\log p)^2$. This strengthens an earlier conjecture of Eswarathasan and Levine that $|J_p|$ is finite for all $p$. Another prediction of the model is that there will be infinitely many pairs $(n,p)$ for which $p^3$ divides the numerator of $H_n$, but only finitely many for which $p^4$ divides $H_n$.

It has been conjectured that there are infinitely many $p$ for which $|J_p| = 3$. We give a probabilistic argument that suggests that such primes have a density $1/e$ in the set of all primes, and experimentally confirm this by a determination of all such $p \le 10^5$.

We investigate the effects of round-off errors on quasi-periodic motions in a linear symplectic planar map. By discretizing coordinates uniformly we transform this map into a permutation of $\Z^2$, and study motions near infinity, which correspond to a fine discretization. We provide numerical evidence that all orbits are periodic and that the average order of the period grows linearly with the amplitude. The discretization induces fluctuations of the invariant of the continuum system. We investigate the associated transport process for time scales shorter than the period, and we provide numerical evidence that the limiting behaviour is a random walk where the step size is modulated by a quasi-periodic function. For this stochastic process we compute the transport coefficients explicitly, by constructing their generating function. These results afford a probabilistic description of motions on a classical invariant torus.

We study self-affine tilings of $\R^n$ with special emphasis on the two-digit case. We prove that in this case the tile is connected and, if $n \le 3$, is a lattice-tile.

Starting from a conjecture of Mazur concerning the topology of rational points on an algebraic variety, we consider the density properties of finitely generated subgroups of rational points on a commutative algebraic group over a number field. In the special case of linear algebraic groups, this problem reduces to a well known conjecture on the algebraic independence of linearly independent logarithms of algebraic numbers.

Partant d'une conjecture de Mazur concernant la topologie des points rationnels sur une variété algébrique, nous étudions la densité des sous-groupes de type fini du groupe des points rationnels d'un groupe algébrique commutatif sur un corps de nombres. Dans le cas particulier des groupes linéaires, le problème considéré se ramène à une conjecture bien connue sur l'indépendance algébrique de logarithmes linéairement indépendants de nombres algébriques.