Integer points and their orthogonal lattices
arXiv:1502.04209 · doi:10.1007/s00222-016-0655-7
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.
15 pages, and 6 pages of appendix, (Appendix by Ruixiang Zhang)
References in corpus (5)
- Distribution of periodic torus orbits and Duke's theorem for cubic fields
- The asymptotic distribution of Frobenius numbers
- Local-global principles for representations of quadratic forms
- Equidistribution of primitive rational points on expanding horospheres
- Integer points on spheres and their orthogonal grids