Counting lattice points and weak admissibility of a lattice and its dual
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COUNTING LATTICE POINTS AND WEAK ADMISSIBILITY OF A LATTICE AND ITS DUAL
BY
Niclas Technau∗ Institut f¨ ur Analysis und Zahlentheorie, Technische Universit¨ at Graz Steyrergasse 30, Graz A-8010, Austria e-mail: [email protected] AND
Martin Widmer Department of Mathematics, Royal Holloway University of London, TW20 0EX Egham, UK e-mail: [email protected]
ABSTRACT
We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box. The error term is expressed in terms of a certain function ν(Γ⊥ , ·) of the dual lattice Γ⊥ , and we carefully analyse the relation of this quantity with ν(Γ, ·). In particular, we show that ν(Γ⊥ , ·) = ν(Γ, ·) for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. This result relies on Beresnevich’s recent breakthrough on Davenport’s problem regarding badly approximable points on submanifolds of Rn . Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.
∗ The first author was supported by the Austrian Science Fund (FWF): W1230
Doctoral Program “Discrete Mathematics”. Received August 17, 2018 and in revised form September 3, 2019
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N. TECHNAU AND M. WIDMER
Isr. J. Math.
Introduction In the present article, we are mainly concerned with three objectives. Firstly, we prove a counting result for lattice points of unimodular weakly admissible lattices in inhomogeneously expanding, aligned boxes. A similar result for homogeneously expanding boxes was proven by Skriganov [22, Thm. 6.1] in 1998. Secondly, we carefully investigate the relation between ν(Γ, ·) (see (0.1) for the definition) and ν(Γ⊥ , ·) of the dual lattice Γ⊥ which captures the dependency on the lattice in these error terms. And thirdly, we apply our counting result to count Diophantine approximations. To state our first result, we need to introduce some notation. By writing f g (or f g) for functions f, g, we mean that there is a constant c > 0 such that f (x) ≤ cg(x) (or cf (x) ≥ g(x)) holds for all admissible values of x; if the implied constant depends on certain parameters, then this dependency will be indicated by an appropriate subscript. Let Γ ⊆ Rn be a unimodular lattice, and let Γ⊥ := {w ∈ Rn : v, w ∈ Z ∀v∈Γ } be its dual lattice with respect to the standard inner product ·, · . Let γn 1/2 denote the Hermite constant, and for ρ > γn set (0.1)
ν(Γ, ρ) := min{|x1 · · · xn | : x := (x1 , . . . , xn )T ∈ Γ, 0 < x 2 < ρ}
where · 2 denotes the Euclidean norm. We say Γ is weakly admissible 1/2 if ν(Γ, ρ) > 0 for all ρ > γn . Note that this happens if and only if Γ has trivial intersection with every coordinate subspace. It is also worthwhile mentioning that the function ν(Γ, ρ) controls the rate of escape of the lattice Γ under the action of the diagonal subgroup of SLn (R) (cf. (1.6)). Furthermore, let T := di
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