Uri Shapira

Stable lattices and the diagonal group

Inspired by work of McMullen, we show that any orbit for the action of the diagonal group on the space of lattices, accumulates on a stable lattice. We use this to settle a conjecture of Ramharter about Mordells constant, get new proofs of Minkowskis conjecture in dimensions up to seven, and answer a question of Harder on the volume of stable lattices.

On the evolution of continued fractions in a fixed quadratic field

We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the ‘normal’ statistics given by the Gauss-Kuzmin measure. As a byproduct, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer k and a quadratic irrational α, the length of the period of the continued fraction expansion of knα equals ckn + o(k15n/16) for some positive constant c. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces.

Integer points on spheres and their orthogonal lattices

Linnik proved in the late 1950’s the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition.

Grids with dense values

Given a continuous function from Euclidean space to the real line, we analyze (under some natural assumption on the function), the set of values it takes on translates of lattices. Our results are of the flavor: For almost any translate, the set of values is dense in the set of possible values. The results are then applied to a variety of concrete examples, obtaining new information in classical discussions in different areas in mathematics; in particular, Minkowskis conjecture regarding products of inhomogeneous forms and inhomogeneous Diophantine approximations.

Homogeneous orbit closures and applications

We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former, for instance we show that if x is the cubic root of 2 then for any y,z in R liminf |n| =0 (as |n| goes to infinity), where denotes the distance of a real number c to the integers.

Measure theoretical entropy of covers

AbstractIn this paper we introduce three notions of measure theoretical entropy of a measurable cover

Equidistribution of divergent orbits and continued fraction expansion of rationals

\mathcal{U}

Equidistribution of divergent orbits of the diagonal group in the space of lattices

in a measure theoretical dynamical system. Two of them were already introduced in [R] and the new one is defined only in the ergodic case. We then prove that these three notions coincide, thus answering a question posed in [R], and recover a variational inequality (proved in [GW]) and a proof of the classical variational principle based on a comparison between the entropies of covers and partitions.