Equidistribution of primitive vectors, and the shortest solutions to their GCD equations

Horesh, Tal; Karasik, Yakov

doi:10.48550/arxiv.1903.01560

Cited by 3 publications

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“…Showing that the image {r(Ω (S,W ) T (Ξ))} T >0 under r is Lipschitz well rounded with Lipschitz constant and T 0 that do not depend on S, W , would imply, according to Corollaries 4.3 in [HK20] and 10.5 in [HK19], that Ω (S,W ) T (Ξ) is LWR with Lipschits constant that is ≺ Ξ e 2(S+W) and T 0 that does not depend on S, W . Now,…”

Section: Proof Of Technical Counting Propositionmentioning

confidence: 99%

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Equidistribution of primitive lattices in $\mathbb{R}^n$

Horesh¹,

Karasik²

2020

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We count primitive lattices of rank d inside Z n , as their covolume tends to infinity, w.r.t. certain parameters of such lattices. These parameters include, for example, the direction of a lattice, which is the subsapce that it spans; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets that are general enough in the spaces of parameters to conclude joint equidistribution of these parameters.The main novelty is that, in addition to the primitive d-lattices Λ themselves, we also consider their orthogonal lattices Λ ⊥ and the quotients Z n /Λ. We are able to count not only w.r.t. parameters of the primitive lattices Λ, but also w.r.t. the same parameters of Λ ⊥ and Z n /Λ. By doing so, we achieve joint equidistribution of, e.g., the shapes of primitive lattices, and the shapes of their orthogonal lattices. Finally, our asymptotic formulas for the number of primitive lattices includes an explicit error term.

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Section: Proof Of Technical Counting Propositionmentioning

confidence: 99%

“…2. The case of d = n−1 was also obtained in [Mar10], using a dynamical approach, as well as in [HK19]. It was also considered in [AES16b, AES16a, EMSS16, ERW17, Ber19] in a more delicate setting.…”

Section: Introductionmentioning

confidence: 99%

Equidistribution of primitive lattices in $\mathbb{R}^n$

Horesh¹,

Karasik²

2020

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“…This text came to life out from the authors' work [HK19,HK20b,HK20a] on equidistrbution problems in geometry of numbers. In the course of our work we relied significantly on counting lattice point results in semisimple algebraic groups.…”

Section: Introductionmentioning

confidence: 99%

A practical guide to well roundedness

Horesh,

Karasik

2020

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Let G be a semisimple algebraic group. We develop a machinery for manipulation and manufacture of well-rounded families {B T } T >0 ⊂ G as they were defined in a work by A. Gorodnik and A. Nevo. The importance of these types of families is that one can asymptotically count lattice points in them and even obtain an error term. Lattice counting is highly effective for solving asymptotic problems from number theory and the geometry of numbers.The tools we develop are handy especially when the family is given w.r.t. some decomposition of G (e.g. Iwasawa or Cartan) and also when it depends upon a sub-quotients of the form M/H, where M ⊂ G is a submanifold and H < G is a closed subgroup.

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“…The first section of the paper is a collection of general facts on p-adic numbers and arithmetic lattices. Sections 2 and 3 are devoted to the proof of Theorem 1, along the lines of the proof given in [HN16,HK19] for the uniform distribution of the real directions of primitive vectors in the unit sphere. It consists of two stages: the first is a translation of the theorem to a statement about counting lattice points in the group SL 2 (R) × SL 2 (Q p ) (Section 2), and the second is proving the counting statement via a method developed in [GN12] (Section 3).…”

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confidence: 99%

“…. All of the above in rank one real Lie groups; in higher rank, we mention[HK19]. We are unaware of equidistribution results of the Iwasawa components of lattice elements in the S-arithmetic setting.The goal of this final subsection is to prove Proposition 20, i.e.…”

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$p$-adic Directions of Primitive Vectors

Guilloux¹,

Horesh²

2021

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Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive Z 2 points in the p-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple S-arithmetic groups.A primitive vector is an n-tuple (a 1 , . . . , a n ) of co-prime integers, and we let Z n prim

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Equidistribution of primitive vectors, and the shortest solutions to their GCD equations

Cited by 3 publications

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Equidistribution of primitive lattices in $\mathbb{R}^n$

Equidistribution of primitive lattices in $\mathbb{R}^n$

A practical guide to well roundedness

$p$-adic Directions of Primitive Vectors

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