At seventeen the densest power of 1 − ζ moves

A lattice with a rotation of prime order $p$ that fixes no vector lives in $p - 1$ dimensions, for the reason the degrees that name the restriction gives, and when the cyclotomic integers $\mathbb{Z}[\zeta_p]$ have class number one every such lattice is that ring with a positive weight on each pair of complex embeddings. A seven-fold lattice lives on a folded torus found the shapes forming a torus folded by the ring’s Galois automorphism, with the ideals $(1 - \zeta)^k$ as the fold’s fixed points. The eleven-fold and thirteen-fold essays carried the picture to ten and twelve dimensions, and found one thing unchanged across all three primes: the densest of the special members was the second power, $(1 - \zeta)^2$.

Thirteen-fold, and nothing is its own dual ended by doubting that this could last. Two primes remain at which the class number is one and the same construction works unchanged: seventeen, in sixteen dimensions, with a cycle of eight special members, and nineteen, in eighteen, with a cycle of nine. Craig’s lattices, which these powers are, are known to reach their best density at a power that grows with the prime, so the peak must move somewhere. The question was whether it had begun to move by seventeen.

It has. At seventeen and at nineteen the densest special member is the third power. And the reason turns out to be simple enough to state as an inequality, which also says why the second power held for three primes and predicts where the fourth takes over.

The same computation, with the prime as a parameter #

The computation is the thirteen-fold one with every thirteen replaced by a parameter, and before it is trusted at seventeen it is asked to reproduce the three primes already done. At seven, eleven and thirteen it finds the densest member at the second power, with kissing numbers 42, 220 and 390, the earlier essays’ values. Then it runs at seventeen and nineteen.

At seventeen there are eight weights, one for each pair of complex embeddings, where the five-fold lattices had two, and seven free numbers once the scale is removed; the doubled logarithms of the cyclotomic units $\sin(\pi a/17)/\sin(\pi/17)$, $a = 2, \dots, 8$, make the space of shapes a seven-dimensional torus. The Galois automorphism, $\zeta \mapsto \zeta^3$ since three generates the multiplicative group modulo seventeen, acts on that torus by an integer matrix of order eight, and $|\det(I - R)|$ is eight, so the fold has eight fixed points. The step from one ideal to the next has order eight on the torus, and the eight ideals $(1 - \zeta)^0$ to $(1 - \zeta)^7$ are the fixed points. At nineteen the same steps give an eight-dimensional torus, a fold of order nine with nine fixed points, and nine ideals.

One numerical precaution is new. The weights of $(1 - \zeta)^k$ are the numbers $|1 - \zeta^j|^{2k}$, which at $k = 7$ range over four orders of magnitude and spoil the shortest-vector search. Because the step has order $H = (p-1)/2$ on the torus, $(1 - \zeta)^k$ is similar to $(1 - \zeta)^{k - H}$, so every member is computed at whichever representative lies nearer zero. The similarity is not assumed: at one power for each prime both representatives are computed and give the same density to six digits and the same kissing number.

Seventeen and nineteen #

The figure at the head of this essay is the seventeen-fold table. $A_{16}^*$ and $A_{16}$ come first, kissing numbers 34 and 272, densities far below the rest. Then the second power, with 952 shortest vectors and centre density 0.01427, and the third, with 1,088 and 0.02151. The third power packs fifty-one per cent better than the second. The fourth falls back to 0.01264. The remaining three, $(1 - \zeta)^5$ to $(1 - \zeta)^7$, are the duals of the fourth, the third and the second, at 0.01905, 0.01119 and 0.00066.

Nineteen repeats the pattern more strongly. The second power has 1,368 shortest vectors and density 0.01207; the third has 1,938 and 0.02443, twice the second. The fourth, at 0.01713, is between them.

The comparison with the best packings known in these dimensions puts the family in its place. In sixteen dimensions the Barnes–Wall lattice $\Lambda_{16}$ has centre density $1/16$, and the seventeen-fold family’s best reaches thirty-four per cent of it. As in twelve dimensions, a $p$-fold rotation is a strong constraint and the densest lattices of a dimension do not have to carry one; what is computed is the best the constraint allows among these lattices.

Duality, and the rule for a self-dual member #

Taking duals sends $(1 - \zeta)^k$ to a lattice similar to $(1 - \zeta)^{1-k}$, as for the earlier primes, and the computation locates each dual on the torus rather than assuming the formula. At seventeen it pairs zero with one, two with seven, three with six and four with five, and fixes nothing, because $2k \equiv 1$ has no solution modulo eight. At nineteen it pairs zero with one, two with eight, three with seven and four with six, and fixes $(1 - \zeta)^5$, because $2 \cdot 5 = 10 \equiv 1$ modulo nine.

The thirteen-fold essay had corrected a rule along the way: a self-dual special member exists exactly when the cycle has odd length, which is when $p$ leaves a remainder of three on division by four. Seventeen leaves one and nineteen leaves three, and both behave as the rule says. The self-dual member at nineteen is not a dense one, with density 0.00672, a quarter of the third power’s. In six dimensions the self-dual member had been the densest; at eleven it was third of five; at nineteen it is fourth of nine. Self-duality and density, which looked related at the first prime, have nothing to do with each other.

The minima are Craig’s bound #

The tables’ minimum column is the key to everything else. Craig proved in 1976 that in the scaling where $(1 - \zeta)$ is the root lattice $A_{p-1}$ with minimum two, the ideal $(1 - \zeta)^k$ has minimum at least $2k$; in the trace-form scaling used here that is a minimum of at least $2kp$. The computation finds the minimum exactly $2kp$ for every power from one to four, at seventeen, nineteen and twenty-three. The determinant needs no bound at all: the ideal $(1 - \zeta)^k$ has index $p^k$ in the ring, and the ring with the trace form has determinant $p^{p-2}$, so the determinant is $p^{p-2+2k}$, which the computation confirms to the last digit of a logarithm.

With both the minimum and the determinant known, the centre density, the minimum’s quarter raised to half the dimension and divided by the square root of the determinant, becomes a formula:

$$\delta(k) = \left(\frac{k}{2}\right)^{(p-1)/2} p^{\,1/2 - k}.$$

It reproduces every computed density at the powers where the bound is attained, to six digits. The ratio between successive powers is then

$$\frac{\delta(k+1)}{\delta(k)} = \frac{(1 + 1/k)^{(p-1)/2}}{p},$$

a growing function of the prime divided by the prime itself.

Why the second power, and for how long #

The ratio answers the question the thirteen-fold essay left in its title. The second power beats the third exactly while $(3/2)^{(p-1)/2} < p$. At seven the left side is about 3.4 against 7; at eleven, 7.6 against 11; at thirteen, 11.4 against 13, close but still below. At seventeen it is 25.6 against 17, and the third power wins. The second power held for three primes because one and a half raised to half the prime grows exponentially and the prime only linearly, and the exponential needed until seventeen to overtake.

The same inequality, one step on, says when the fourth power takes over: when $(4/3)^{(p-1)/2} > p$. At nineteen that is 13.3 against 19, so the third power keeps the lead. At twenty-three it is 23.7 against 23, barely over. The rule therefore predicts that the fourth power is densest at twenty-three, by a margin of about three per cent. The computation, run directly on the twenty-two-dimensional lattices, finds $(1 - \zeta)^4$ at 0.03510 and $(1 - \zeta)^3$ at 0.03409: the fourth power wins by 2.9 per cent, as predicted.

Twenty-three is outside the family the rest of the essay describes. Its cyclotomic integers have class number three, the first prime at which the class number is not one, so not every twenty-two-dimensional lattice with a twenty-three-fold rotation is the ring with weights, and the torus picture no longer covers them all. The ideals $(1 - \zeta)^k$ are principal and remain perfectly good lattices, and they are the ones the formula speaks about. Further out the rule gives the fourth power at twenty-nine and thirty-one and the fifth at thirty-seven. Those three predictions assume the bound is still attained there, which has not been computed.

The shortest vectors, counted in orbits #

The kissing numbers in the tables have a structure the formula does not supply. Every lattice in the family carries the rotations $\pm\zeta^c$, $2p$ of them, and none of them fixes a non-zero vector, so the shortest vectors of any member fall into orbits of exactly $2p$. Dividing the kissing numbers by $2p$ counts the orbits.

For the first power the count is $H = (p-1)/2$ at every prime: 3, 5, 6, 8, 9 and 11 orbits at 7, 11, 13, 17, 19 and 23. That is the root lattice $A_{p-1}$, whose $p(p-1)$ roots are the differences of the $p$ basis vectors of the larger lattice it sits in, and the count says nothing new. For the second power the count is the number of pairs among the $H$, 3, 10, 15, 28, 36 and 55, at every prime computed. That suggests a description of the shortest vectors of $(1 - \zeta)^2$ as products of two shortest vectors of $A_{p-1}$ taken from different orbits, which would have to be checked rather than assumed: the computation counts the vectors and does not identify them.

The third power has no such count. Its orbits number 1, 5, 12, 32, 51 and 99 at the six primes, which is neither the number of triples among the $H$ nor any simple fraction of it; at seventeen the triples number 56 and the orbits 32. So the formula that gives the density exactly says nothing about how the shortest vectors are arranged, and the arrangement changes character between the second power and the third. That is a second difference between the powers, independent of which packs better, and it is one this essay does not explain.

It also bears on the comparison with the best known packings. The Barnes–Wall lattice’s 4,320 shortest vectors in sixteen dimensions are four times the third power’s 1,088, and its density three times as large. Covering and packing want different lattices is the reminder that the comparison is of one property only, and nothing here measures how well these lattices cover space or how well they serve as codes.

How fast the peak moves #

The inequality also says how the peak travels as the prime grows, provided the bound stays attained. The ratio $(1 + 1/k)^{(p-1)/2}/p$ crosses one where $(p-1)/2 \cdot \log(1 + 1/k) = \log p$, and since $\log(1 + 1/k)$ is close to $1/k$, the densest power sits near

$$k^* \approx \frac{p - 1}{2 \log p}.$$

That expression gives 2.8 at seventeen, 3.1 at nineteen, 3.5 at twenty-three, 4.2 at twenty-nine and 5.0 at thirty-seven, against densest powers of 3, 3, 4, 4 and 5 from the exact inequality. The peak therefore moves about one step for every few primes and keeps moving: in a family of dimension $p - 1$ the best power grows like the dimension divided by its logarithm. That is the growth Craig’s lattices were built to exploit, and it is why they become competitive only in high dimensions, where a large power is possible and the bound’s factor $k/2$, raised to half the dimension, can overcome the determinant’s $p^{k}$.

It also puts the first three primes in proportion. At seven, eleven and thirteen the estimate is 1.5, 2.1 and 2.3, so the second power was not a stable feature of the family but the integer nearest a slowly growing quantity, and the densest lattice in the plane is the far end of the same story, a dimension so low that no power past the first can compete.

Where the bound is not attained #

The formula only speaks for powers whose minimum is exactly $2kp$, and the computation finds powers where it is larger. At eleven the fourth power, at thirteen the fifth, and at seventeen the fifth have minima above the bound, so their densities exceed the formula’s, at seventeen 0.01905 against a formula value of 0.00443. These are the powers past the middle of the cycle, the duals of the low powers, whose minima are set by the dual’s structure rather than by Craig’s bound. None of them overtakes the formula’s winner at any prime computed, so the rule’s prediction of the densest power survives them. But it is a statement checked prime by prime, not a consequence of the formula, and a prime where a power beyond the middle overtook the predicted peak would break the rule’s reach.

The rest of the family, sampled #

The special members are the fold’s fixed points, but the family is the whole torus, and in seven dimensions a grid fine enough to be convincing is out of reach. The two-point grid sampled here, a lattice at the centre of each of 128 cells, finds nothing denser than 0.0143, two thirds of the third power. Most of the family packs far worse, as at thirteen, where finer grids were possible and found the same. That the special members are the family’s densest is supported, not proved.

What the family must refuse #

The refused claim is the one the earlier essays had made three times: that $(1 - \zeta)^2$ is the densest special member. At seventeen and nineteen it is not, and the reason is now a formula rather than a census. Every other test could fail as well. The fold’s fixed points are counted from a determinant and matched member by member, the duals are located on the torus rather than read off a formula, the similarity of a power with its representative near zero is computed both ways, the bound’s attainment is tested at every power from one to four, and the rule’s prediction at twenty-three was made before the twenty-two-dimensional lattices were computed.

Kummer, Craig, and a bound that is exact #

Kummer’s cyclotomic integers supply the ring, and the class number of $\mathbb{Z}[\zeta_p]$ is one for every prime up to nineteen and for no prime beyond, which is why this essay is the last of its line that can describe the whole family. Craig’s lattices of 1976 were introduced as a source of dense packings in dimensions where little better was known, and Conway and Sloane’s book tabulates them. What the computation adds here is the observation that the bound is attained exactly for the low powers at these primes, which turns the question of the peak into the one-line inequality above, and the check of that inequality against lattices actually computed, including one at which it predicted the outcome before the computation was run.

Still open: whether the bound stays exact #

The whole rule rests on the minimum of $(1 - \zeta)^k$ being exactly $2kp$ for the small powers. Craig’s bound is a lower bound, and the computation found it attained for $k \le 4$ at seventeen, nineteen and twenty-three and exceeded beyond the middle of each cycle. Whether it stays attained for the powers near the predicted peak as the prime grows, at twenty-nine, thirty-one and thirty-seven, decides whether the predictions of four, four and five hold. A minimum larger than the bound at the predicted peak would raise that power’s density and could only strengthen it; a minimum larger at the power after it could move the peak further on. The shortest-vector searches would be in twenty-eight, thirty and thirty-six dimensions, harder than anything run here but not out of reach for lattices this symmetric. They have not been run.