Primitive Quartic Torus Packets Equidistribute Across Arbitrary Orders

The manuscript claims that complete arithmetic families of volume-one lattices become equidistributed as the absolute discriminant of the order grows. Take any sequence of primitive totally real quartic fields, any order in each field, and any ordering of the four real embeddings. If the absolute order discriminants tend to infinity, the associated packet measures converge to Haar probability on the space of volume-one lattices. Equivalently, the average of every continuous, compactly supported test function against a packet converges to its Haar average.^†

The claim concerns a specific packet. For a quartic field K, “primitive” means there is no proper intermediate field between the rationals and K. Choose an order O, a full-rank subring containing 1, and take all ordinary classes of proper invertible fractional O-ideals. Embed each ideal through the four real embeddings of K, normalize its covolume to one, and let positive diagonal stretching generate a compact three-dimensional torus orbit. Include all 16 coordinate sign patterns, count each distinct orbit once, and weight each orbit by invariant volume before normalizing by the packet’s total volume.

The theorem also asserts no escape of mass: for any ε > 0, some compact region of the lattice space contains at least 1 − ε of every sufficiently late packet. This matters because a volume-one lattice can become arbitrarily thin: one nonzero vector shrinks while another direction expands, taking the lattice outside compact regions.

The proof first bounds how often a packet contains a short vector. In the large-discriminant limit, the probability of a nonzero vector with sup norm below r is at most Cr⁴. This gives tightness, so limiting measures do not lose mass at infinity. A second vector estimate controls small neighborhoods; combined with diagonal dynamics, it yields positive entropy in almost every ergodic component of a limit.

Positive entropy does not by itself force the limit to be Haar. The measure-rigidity argument leaves a specific proper alternative in dimension four: a homogeneous component with two coordinate blocks of size two. Primitivity of the original fields does not automatically rule out this structure in a limit. The proof must therefore show that these “two-plus-two” components have zero limiting mass.

The two-plus-two obstruction can be tracked in the exterior square of a lattice, which has six coordinates, one for each pair of coordinate directions. The point of this calculation is to translate the obstruction’s four small coordinates into a small region of invariant data that can be counted arithmetically.

Write the coordinates as v₁₂, v₁₃, v₁₄, v₂₃, v₂₄, and v₃₄. Under diagonal stretching by factors a₁, a₂, a₃, and a₄, coordinate vᵢⱼ is multiplied by aᵢaⱼ, where a₁a₂a₃a₄ = 1. Thus the products y₁ = v₁₂v₃₄, y₂ = −v₁₃v₂₄, and y₃ = v₁₄v₂₃ are invariant: each acquires all four stretching factors, whose product is one.

For an exterior lattice vector in a volume-one lattice, their sum Q(v) = y₁ + y₂ + y₃ is an integer. The vector need not be a single decomposable wedge; the example v = e₁ ∧ e₂ + e₃ ∧ e₄ has Q(v) = 1.

Now consider an obstruction supported on the opposite pair 12 and 34, with both coordinates nonzero. The other four coordinates are small: if each has absolute value below r, then |y₂| = |v₁₃v₂₄| < r² and |y₃| = |v₁₄v₂₃| < r². This condition does not itself rule out the obstruction. Instead, it confines its invariant data to a small region. Fix the nonzero integer m = Q(v); then y₁ = m − y₂ − y₃, so the possible (y₂, y₃) values lie in a square of side 2r², with area 4r⁴. Halving r shrinks that area by a factor of 16. The same construction applies to the other two coordinate pairings.

The geometric square alone does not count arithmetic vectors. The manuscript places the three products into the three real embeddings of a totally real cubic resolvent field. Their integer sum becomes a fixed nonzero trace. A central counting theorem bounds the weighted number of such products in a rectangle by a constant times the rectangle’s area and an arithmetic normalization. The weights account for how many exterior vectors produce the same products and cannot simply be dropped. Both raw traces, m and −m, are retained.

There is a scale restriction. Fix the nonzero trace and a bounded region first; the rectangle’s side lengths must be at least X to the power −τ, for some τ > 0, where X = q√D = √|Disc(O)|, q = [O_K : O], and D = |Disc(K)|. This is not an unrestricted estimate at arbitrarily small scales.

Uniformity over arbitrary orders is a central difficulty. Since |Disc(O)| = q²D, the order index can grow even when the field stays fixed. Primes dividing the field discriminant or the order index can distort local counting weights. The proof retains those weights, studies their quadratic Fourier transforms, and uses Poisson summation; a weighted sieve handles the remaining primes. A stabilizer order in the cubic field records the integral exterior lattice, with its discriminant growing with the original order’s complexity. These are the technical steps behind the rectangle bound; the explainer gives their roadmap, not the estimates themselves.

The paper integrates the vector counts along diagonal orbits. Logarithmic fiber lengths enter, but a summable subdivision preserves the fourth-power bound. For a fixed nonzero trace and a fixed bound on the two potentially large coordinates, the argument first takes a packet subsequential limit and then lets r shrink to zero. The obstruction consequently has zero limiting mass. A countable union over traces and coordinate bounds, and over the three coordinate pairings, excludes every proper alternative.

With no escape of mass and no surviving two-plus-two component, Haar probability is the only remaining limit. The conclusion is equidistribution of complete, volume-weighted packets as the order discriminant grows—not a claim about selected ideals or individual torus orbits.