Counting points of a discrete subgroup near a dilated Lipschitz-parametrizable set

Let L be a discrete additive subgroup of a proper normed real vector space E, and let S ⊆ E be Lipschitz parametrizable in dimension d, that is, covered by finitely many Lipschitz images of the unit d-cube. Dilating S by a factor c ≥ 1 and thickening it by a fixed bounded set B produces a region that carries O(c ^ d) points of L.

This is the quantitative half of Lipschitz parametrizability. When d is strictly smaller than the ambient dimension, as in the codimension-one boundary application, it is a genuinely smaller order than the c ^ (dim E) points carried by a dilated body and hence gives a power-saving error term in a lattice-point count. The thickening by B is what the application needs: the lattice cells x + F that meet a dilated region c • S are exactly the x ∈ L lying in c • S + (-F), so a count of cells meeting the boundary of a dilated body is a count of the points of L in such a region.

The proof subdivides the unit cube into m ^ d subcubes of side 1 / m, with m of size c, so that each chart maps a subcube into a set of diameter at most one after dilating by c. Translation invariance bounds the number of points of L in any set of bounded diameter by a constant, so the total count is at most a constant times the number m ^ d of subcubes.

Main results

• TauCeti.IsLipschitzParametrizable.finite_smul_add_inter: a bounded thickening of a dilated Lipschitz-parametrizable set meets a discrete subgroup in a finite set.
• TauCeti.IsLipschitzParametrizable.exists_ncard_smul_add_inter_le: the explicit bound #((c • S + B) ∩ L) ≤ A * c ^ d for c ≥ 1, with A independent of c.
• TauCeti.IsLipschitzParametrizable.isBigO_ncard_smul_add_inter: the same bound as an asymptotic statement, #((c • S + B) ∩ L) = O(c ^ d) as c → ∞.

References

• S. Lang, Algebraic Number Theory, Chapter VI, Section 2.

theorem TauCeti.IsLipschitzParametrizable.finite_smul_add_inter {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] {d : ℕ} {S : Set E} (hS : IsLipschitzParametrizable d S) (L : AddSubgroup E) [DiscreteTopology ↥L] {B : Set E} (hB : Bornology.IsBounded B) (c : ℝ) : ((c • S + B) ∩ ↑L).Finite

A bounded thickening of a dilated Lipschitz-parametrizable set meets a discrete subgroup in a finite set.

theorem TauCeti.IsLipschitzParametrizable.exists_ncard_smul_add_inter_le {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] {d : ℕ} {S : Set E} (hS : IsLipschitzParametrizable d S) (L : AddSubgroup E) [DiscreteTopology ↥L] {B : Set E} (hB : Bornology.IsBounded B) : ∃ A ≥ 0, ∀ (c : ℝ), 1 ≤ c → ↑((c • S + B) ∩ ↑L).ncard ≤ A * c ^ d

The boundary count. If S is Lipschitz parametrizable in dimension d and B is bounded, then for c ≥ 1 the thickened dilate c • S + B contains at most A * c ^ d points of a discrete subgroup L, with the constant A independent of c.

With B = -F for a bounded fundamental domain F of L, the left-hand side counts the cells of L that meet c • S; taking S to be the frontier of a body and d its codimension-one parametrization dimension is what produces a power-saving error in a lattice-point count.

theorem TauCeti.IsLipschitzParametrizable.isBigO_ncard_smul_add_inter {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] {d : ℕ} {S : Set E} (hS : IsLipschitzParametrizable d S) (L : AddSubgroup E) [DiscreteTopology ↥L] {B : Set E} (hB : Bornology.IsBounded B) : (fun (c : ℝ) => ↑((c • S + B) ∩ ↑L).ncard) =O[Filter.atTop] fun (c : ℝ) => c ^ d

The boundary count as an asymptotic statement: if S is Lipschitz parametrizable in dimension d and B is bounded, the number of points of a discrete subgroup L in the thickened dilate c • S + B is O(c ^ d) as c → ∞.