The Pythagorean Angle Lattice and Minkowski's Question-Mark Function: Rényi Spectrum and Extremal Local Dimensions

In the Minkowski coordinate the PAL forest law is exact dimensional of dimension 3 log 3/(5 log 2), admits an explicit Rényi spectrum, and attains extremal local dimensions log 3/(2 log 2) and 3 log 3/(4 log 2) at the boundary and at √2 − 1. Let U be the group of rational points of the unit circle and 𝒜 = {Arg z : z ∈ U \ {±1, ±i}}. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}; its angles are the arguments Arg(W²) of primitive Gaussian integers W. The primitive W = a + bi with a, b > 0 and odd norm carry the lattice angles θ_W = 2 arg W, and contracting the Calkin–Wilf tree to them gives a forest of two ternary trees whose uniform generation laws converge to a stationary law. We study this law in the Minkowski coordinate ?(b/(a + b)), which is common to all depth layers of the lattice. Its image η on [0,1] is given by an explicit three-state automaton with weighted permutation matrices. We prove that the binary digit process of η has no finite Markov order, that η is exact dimensional with dim_H η = 3 log 3 / (5 log 2), and that the forest law is singular with respect to Minkowski’s measure. For every real q > 0 the dyadic Rényi dimension is D_q(η) = 1/(q − 1) · log₂ (√(1 + 8 · 3^q) − 1)/4, in particular D₂(η) = log₂ (√73 − 1)/4. The lower and upper local dimensions of every point lie in [log 3/(2 log 2), 3 log 3/(4 log 2)], every exponent of this interval is the local dimension of some point, and both ends are attained, the upper one at the irrational slope √2 − 1 of angle θ = π/4, and both corresponding level sets have Hausdorff dimension 1/2. The cylinders of minimal mass at level 4m are counted by the central binomial coefficients, the Lebesgue-typical local dimension is 2 log 3/(3 log 2). For negative orders, without assuming that the pressure limit exists, we bound the lower and upper exponential growth rates of the partition sums explicitly; both strictly exceed the rate of the exactly solvable 00/11 subsystem, and the excess tends to 0 as the order tends to −∞.