The set of primitive vectors on large spheres in the euclidean space of dimension d 3 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector, together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics, we obtained the desired equidistribution result in dimension d 6 and in dimension d = 4, 5 under a mild congruence condition on the square of the radius. The case of d = 3 is considered in a separate paper.
Abstract. A finitely generated subgroup Γ of a real Lie group G is said to be Diophantine if there is β > 0 such that non-trivial elements in the word ball BΓ(n) centered at 1 ∈ Γ never approach the identity of G closer than |BΓ(n)| −β . A Lie group G is said to be Diophantine if for every k ≥ 1 a random k-tuple in G generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non Diophantine nilpotent and solvable (non nilpotent) Lie groups.
Abstract. We study the general problem of extremality for metric diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over Q, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest. In particular we prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.
Abstract. 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.
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