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.
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1] 2 , possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod 1} n∈N are uniformly eventually bounded.
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.
We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood's conjecture:where · denotes the distance from the nearest integer. The existence of even a single pair that satisfies statement (C1), solves a problem of Cassels from the 50's. We then prove that if 1, α, β span a totally real cubic number field, then α, β, satisfy (C1). This generalizes a result of Cassels and Swinnerton-Dyer, which says that such pairs satisfy Littlewood's conjecture. It is further shown that if α, β are any two real numbers, such that 1, α, β, are linearly dependent over Q, they cannot satisfy (C1). The results are then applied to give examples of irregular orbit closures of the diagonal group of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity.
Topology on PM(X)In this section, we study the topology τ P on PM(X) for any locally compact second countable Hausdorff space X. We will give a description of
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