Matching gaps in the Pythagorean angle lattice: exact gaps, density and the lifetime spectrum

Matching gaps of the Pythagorean Angle Lattice are parametrized by roots of s² ≡ −1 mod 2m, obey an exact lifetime spectrum in ℤ[√2], and have asymptotic density (3 − 2√2)Q/(2π) by Hooley’s theorem. Let 𝒰 be the group of rational points of the unit circle and μ₄ = {±1, ±i} its torsion. The Pythagorean angle lattice is PAL = {cot(θ/D) : θ ∈ 𝒜, D ≥ 1}, where 𝒜 = {Arg z : z ∈ 𝒰 \ μ₄}; its angles are the angles Arg(W²) of squares of primitive Gaussian integers. We study the depth-two directions ½ Arg z of the points z ∈ 𝒰 of height at most Q, taken modulo the torsion rotation, and the gaps between consecutive ones. Consecutive directions have determinant 1 or 2, the largest gap is exactly arctan(1/(2B(Q)+1)) with B(Q) = ⌊(√(2Q−1)−1)/2⌋, and the two gaps next to a direction of height h are 1/√(2hQ) + O(1/Q). A gap whose two endpoints come from Gaussian integers of odd norm is called a matching gap. Matching gaps are in bijection with the roots of s² ≡ −1 (mod 2m) in the window 2m² − 1 < s² < 4m² − 1; a gap is born at height m, dies at height (s² + 1)/(2m), and its lifetime is always divisible by 4. Using Hooley’s equidistribution theorem for the roots of 2z² + 2z + 1, we prove that the number of matching gaps alive at height Q is ((3 − 2√2)/(2π)) Q + o(Q), and we compute the birth and death rates and the contribution of each generation. The gaps of a fixed deficit δ number 2^{ω(δ)} ((√2 − 1)/(2√δ)) √Q + O(2^{ω(δ)}), and every gap alive at Q was born at height at least Q/2 + 1 + √Q. A lifetime L = 4t occurs if and only if 8t² + 1 is a norm from ℤ[√2], and then the gaps of lifetime L form exactly a(8t² + 1) chains under a half-step map, where a(N) is the number of ideals of norm N in ℤ[√2]. The mean of a(8t² + 1) is (2√2 log(1 + √2))/π, the set of lifetimes up to X has between X^{1−o(1)} and O(X/√(log X)) elements, the least prime obstructing a lifetime 4t equals p for a proportion (2/(p+1)) ∏_{q