Integer points on spheres and their orthogonal lattices

Abstract

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.

Joinings of higher rank torus actions on homogeneous spaces

“…There seems to be substantial scope for applying the theorems presented above to arithmetic questions. A concrete result in this direction by M. Aka, U. Shapira and the first named author [1] considers for all integer points (x, y, z) of the sphere x 2 + y 2 + z 2 = m both the normalized point m −1/2 (x, y, z) on the unit sphere S 2 and the shape of the lattice orthogonal to (x, y, z) in Z 3 , which corresponds to a point in PGL(2, Z)\H. Using Theorem 1.4 it is shown that these collections of pairs in S 2 ×PGL(2, Z)\H become equidistributed as m → ∞ along squarefree 1 integers, satisfying both m ≡ 0, 4, 7 mod 8 (a necessary and sufficient condition for existence of primitive integral points on the sphere of radius √ m) as well as an auxiliary condition of Linnik type-namely that (−m) is a quadratic residue modulo two distinct fixed primes.…”

Equidistribution of rational subspaces and their shapes

“…There exists an analogous conjecture for k = 1, n − k ≥ 2 where one only considers the pairs (L, [L ⊥ (Z)]) (and similarly for n − k = 1, k ≥ 2). This has been studied extensively by the first named author with Einsiedler and Shapira in [AES16b,AES16a] where the conjecture is settled for n ≥ 6 (i.e. n − k ≥ 5), for n = 4, 5 under a weak congruence condition and for n = 3 under a stronger congruence condition on D. We remark that, as it is written, [AES16b,AES16a] treat only the case where Q is the sum of squares (that we will sometimes call the standard form), but the arguments carry over without major difficulties.…”

“…This has been studied extensively by the first named author with Einsiedler and Shapira in [AES16b,AES16a] where the conjecture is settled for n ≥ 6 (i.e. n − k ≥ 5), for n = 4, 5 under a weak congruence condition and for n = 3 under a stronger congruence condition on D. We remark that, as it is written, [AES16b,AES16a] treat only the case where Q is the sum of squares (that we will sometimes call the standard form), but the arguments carry over without major difficulties. Using effective methods from homogeneous dynamics, Einsiedler, Rühr and Wirth [ERW19] proved an effective version of the conjecture when n = 4, 5 removing in particular all congruence conditions.…”

“…The case n = 3 relies on a deep classification theorem for joinings by Einsiedler and Lindenstrauss [EL19]; effective versions of that theorem are well out of reach of current methods from homogeneous dynamics. Assuming the Generalized Riemann Hypothesis, Blomer and Brumley [BB20] have recently removed the congruence condition in [AES16b].…”

Linnik’s problems and maximal entropy methods

“…The present article aims to give an ergodic theoretic proof of Linnik's theorem using maximal entropy methods and following Einsiedler, Lindenstrauss, Michel and Venkatesh [11]. The motivation for such a proof originates from a refinement of Linnik's theorem by Aka, Einsiedler and Shapira in [3].…”