Interactive companion · residue sequences
Page 1 located the failure of the classical sequences in one place: each coordinate reads the index on its own, so a carry in one coordinate is invisible to the others. The repair has to tie the coordinates together, with one fixed scramble that serves every number of points. A classical identity, the residue theorem (familiar from complex analysis, and valid for rational functions over any field), does exactly that.
All points are computed in your browser from the closed-form generator matrices of the paper (def:residuestar, def:residue; block-periodic matrices exported from the audited builder). Separations are exact and toroidal; certified bounds are truncated, measured values rounded.
A Taylor digit of the index polynomial V at a point a is a residue: the coefficient of (z − a)n in V equals the coefficient of (z − a)−1 in V·(z − a)−n−1. Every scramble of a Faure-type sequence can be written this way: digit n of coordinate c is the residue at the coordinate's own point ac of V times a function with a pole of order n + 1 there (lem:resform).
The residue theorem says that the residues of one rational function at all its poles add up to zero (when it falls off fast enough at infinity). So if every coordinate reads its digits from one common rational function, the theorem is a linear identity between digits of different coordinates. It holds at every level, and it involves no carries at all: a carry in one coordinate is now seen by the others.
The simplest example is the brick wall W⁻ in dimension two (ex:brickwall). Its identity says: the height of a point inside its cell (the first digit of coordinate 1 below the cell) is half the finest cell digit of coordinate 0, modulo p, at every level. Neighbouring columns of points are staggered by half a cell, like the courses of a brick wall. Click a column to read it.
Where does the half come from? The brick wall reads coordinate 0 through 1/((z − 1)(w/θ)n+1) at the point 0 and coordinate 1 through 1/((z − 1)(w/θ)n) at the point 1, with w = z(z − 1), θ = −1/4 and constant factors −4 and 2. For one function F both residues appear, they add up to zero, and the constants turn −4 and 2 into the factor ½. The proof of the bound uses only this one coordinate: a step to the next column moves the height by exactly half a cell whatever the carry, so neighbours are at least (p − 3)/(2p) apart (thm:WF). Coordinate 0 is left uncontrolled, and the closest pair sits in neighbouring columns at almost the same horizontal position.
The brick wall reads its identity at two scales: coordinate 0 at the scale of the cells, coordinate 1 one digit deeper. That is why coordinate 0 keeps a jitter of a whole cell. The remedy is to read the identity at one scale in all coordinates: let every coordinate read the same function V dz, divided by a constant vc. Then the residue theorem ties together the last cell digits of all coordinates, at the prefixes PsK−1 of psK−1 points, one digit before each full level:
v0 y0,K−1 + v1 y1,K−1 + … + vs−1 ys−1,K−1 ≡ 0 (mod p)
This is the skeleton relation (lem:residue-rel), and v is its check vector. Read it with K digits in every coordinate: the cubes of side p−K, the fine cells, are p times as many as the points, and the occupied ones are exactly those whose index b satisfies v·b ≡ 0 (mod p), one point in each (thm:residue). These indices form the lattice Λv of coding theory's Construction A (Conway and Sloane, Ch. 5). The points are therefore a lattice (the corners of their fine cells) plus a jitter of less than one fine cell, and two points are at least the shortest lattice vector minus one unit of jitter per coordinate apart (the quantity λjit of the companion paper):
sep(PsK−1) ≥ p−1/s λjit(Λv), λjit(Λ) = min over b ≠ 0 in Λ of ‖(|b| − 1)+‖, for every K.
Choose a base and a check vector v = (x, −1). The zoom shows 13 × 13 fine cells around the closest pair, the cells of Λv shaded and two short lattice vectors drawn.
Skeleton coordinates. Another way to read the same relation: give one coordinate only K − 1 cell digits and the others K. Those boxes are exactly as many as the points, so each holds one, and the next digit of the short coordinate (its skeleton digit) is fixed by the relation, a linear rule in the last cell digits of the others. This is the picture of the nets of the companion paper (a lattice image plus jitter, with one skeleton coordinate, in its language), with one difference that matters here: those nets are built for one size at a time, while one fixed scramble now gives the lattice at every size psK−1 at once, because the residue theorem does not care about K.
At the full levels in between, the residue sequence has only the inclusion bound λjit/p, from PsK ⊂ Ps(K+1)−1 (rem:residuefull); the panel shows the measured value there as well. That is a statement about this bound only; page 3 explains why one exact relation per level is about all a fixed scramble can carry.
The construction is the same in every dimension. In dimension three with p = 11 and the check vector v = (9, 4, −1), the prefixes P2 (121 points) and P5 (161 051 points) occupy exactly the fine cells of Λv. Left: one layer of fine cells, perpendicular to the third axis; within a layer the occupied cells form a shifted copy of a planar lattice, and the shift moves with the layer. Right: a block of fine cells around the closest pair; drag to turn it.
The bound depends only on the lattice Λv. The Korobov check vector (Korobov; Sloan and Joe) v = (ρ, ρ2, …, ρs−1, −1), ρ = ⌊p1/s⌋, has λjit ≥ ρ − 1, which gives in every dimension and for every prime p > 2s
sep(PsK−1) ≥ (⌊p1/s⌋ − 1) p−1/s > 1 − 2p−1/s for every K (cor:korobov).
Searched check vectors do better, and exceed 1, the value of the square grid, already at moderate primes: a lattice can pack better than the grid (the hexagonal lattice reaches 1.0746 in the plane). Sakai and Goda also choose Korobov generating vectors for separation, by lattice basis reduction, for lattice point sets, one for each N (Sakai and Goda). Compute any case below; "search all" reproduces the exhaustive rows of tab:residue in your browser. For p ≤ 2s no check vector has λjit > 0 (cor:residueQU), and whether residue sequences are quasi-uniform there is not addressed by the paper.
Certified bounds of tab:residue (truncated). E: exhaustive over all check vectors; R: random search over 3·106 vectors, a lower bound on the best possible. The best values appear to approach √γs, the square root of the Hermite constant (1.122, 1.189, 1.231 for s = 3, 4, 5); that limit is open.
The brick wall generalizes too. In the staggered residue sequence one coordinate c⋆ reads one digit deeper than the others, and its geometric digit at every full level is a rank-1 lattice rule in the cell indices of the others (lem:residuestar-rel):
rc⋆ ≡ k1 n1 + … + ks−1 ns−1 (mod p).
A step in another coordinate moves the staggered coordinate by a fixed amount whatever the carry, so this certifies every full level. With the dyadic weights k = (21−s, …, 1/4, 1/2) the bound is (⌊p/2s−1⌋ − 1)/p for every prime p > 2s, and no weights do better for this bound (cor:dyadic): the other coordinates are not controlled inside their cells, so only one coordinate's gap is counted, and 2s−1 positions must share it. The cap belongs to the certificate, not the sequence: the exact values of sep(Ps) computed in dimensions four and five lie between 0.1985 and 0.4837.
Certified constants for every K (tab:residuecmp, truncated): staggered residue sequence at every full level, residue sequence at m ≡ −1 (mod s), and the bound for every N that each gives through lem:prefix, LB/p in units of 10−3.
The nets of the companion paper reach separations 1 − O(1/q) by prescribing all s geometric digits of a full level by a fixed linear rule in the last few cell digits, a design. One residue identity fixes one digit. To fix s of them, the block-periodic residue sequence reads the digits of each coordinate in blocks of s, with s different factors in front of the differential; the residue theorem then installs the complete design at every s-th full level, N = ps2K (thm:block). In dimension two with the design F0 = (−1 1; −1 −1) this gives sep ≥ 1 − 2/p at N = p4K for every prime p ≥ 3; in dimension three 1 − 2/p at p9K for p ≥ 5.
The picture shows 11 × 11 cells around the closest pair. At a certified level the lattice images (open circles) form patches of the slightly rotated square lattice (I + F0/p)ℤ2 (light segments), separated by slip planes where a geometric digit wraps around. At a full level outside the family nothing is designed.
These are the strongest constants of the paper at their certified sizes, from small primes on. For every N they are weaker: the certified sizes are s2 digits apart, and the bound for all N (lem:prefix) loses the factor p−s instead of p−1: for s = 2, p = 5, 0.024 against 0.089 for the residue sequence. With a longer block the same scramble also carries skeleton relations at most levels in between (thm:blocktwo). Why not the full design at every full level? Page 3.
Dimensions four to seven: the free diagonal. For s ≥ 4 the design is the one of the nets, U*s, moved to the places 0, 1, ½, …, 1/(s − 1). Its diagonal is chosen for diagonal dominance, which the nets need: on every step every moving coordinate moves backward. That costs the axis step along a coordinate a*s + 1 units, so the bound is 1 − (a*s + 1)/p. The sequence never uses the diagonal (only the entries off the diagonal must be reachable), so it can be lowered as long as every step keeps one short backward coordinate or two backward ones (lem:signclass, a finite check). The chart shows the certified constant LB(F) at the primes where the paper states it, for both diagonals; each value is an exact evaluation of the one-digit relaxation.
Companion to L. Gruenschloss, Low-discrepancy quasi-uniform digital sequences in every dimension (draft v4, 2026), cited by its \label names. Results without a credit are from that paper. Where these pages call a statement new, it means: not found in the literature we searched (through October 2026).
On this page: in genus 0 the residue-form generator matrices are exactly the construction of Mayor and Niederreiter, which contains every sequence of the paper; Niederreiter and Özbudak used the residue theorem across places for a duality that bounds t. New in the sense above are the choice of the differentials that makes the residue theorem give one digit of each coordinate, and the use of these relations for the separation. The Korobov check vector has the form of the generating vectors of Korobov's lattice rules (Korobov; Sloan and Joe), and Λv is a Construction A lattice (Conway and Sloane); the lattice image, jitter, λjit, skeleton coordinates and designs come from the companion nets paper.