INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A28

PERIODIC MULTIPLICATIVE ALGORITHMS OF SELMER TYPE

Fritz Schweiger

Department of Mathematics, University of Salzburg, Hellbrunnerstrasse 34 A-5020 Salzburg / Austria

                         fritz.schweiger@sbg.ac.at

Received: 2/1/05, Revised: 11/29/05, Accepted: 11/29/05, Published: 11/30/05

Abstract

The multiplicative acceleration of the 2-dimensional Selmer algorithm is considered. Its behavior is more or less unknown and no general result on convergence is known. In this paper we show that periodic expansions do in fact converge and the coordinates of the limit point are rational functions of the largest eigenvalue of the periodicity matrix. Some comparisons are made with the Jacobi-Perron algorithm (Remark1) and the triangle sequence (Section 3).

Keywords: Multidimensional continued fractions, periodic expansions Mathematics Subject Classification (2000): 11K55, 11J13, 11J70, 28D05.

1. Introduction

Recently a renewed interest on multidimensional continued fractions has been observed. A survey of recent literature is given in Schweiger ([5],[6]).

Special attention has been paid to 2-dimensional algorithms. The most well known example is the Jacobi-Perron algorithm. Choose integers b_{0}, b_{1}, b_{0} such that b_{0} ≥ b_{1} > 0, b_{0} ≥ b_{2} > 0 and define

k_{1} := [\frac{b_{2}}{b_{1}} ], k_{2} := [\frac{b_{0}}{b_{1}} ]

and

κ(b_{0}, b_{1}, b_{2}) := (b_{1}, b_{2} − k_{1}b_{1}, b_{0} − k_{2}b_{1}).

This can be transformed into a 2-dimensional map as follows. We use the projection

π(b_{0}, b_{1}, b_{2}) := (\frac{b_{1}}{b_{0}} , \frac{b_{2}}{b_{0}} ) .

Then there is a unique map K such that π ∘ κ = K ∘ π. The map K is given piecewise on a suitable subset of the unit square as

K(x_{1}, x_{2}) = (\frac{x_{2}}{x_{1}} − k_{1}, \frac{1}{x_{1}} − k_{2})

and is therefore an obvious generalization of the map connected with regular continued fractions.

Selmer [7] proposed the following algorithm. Choose integers b_{0} ≥ b_{1} ≥ b_{2} > 0 and define

σ(b_{0}, b_{1}, b_{2}) := (b_{0} − b_{2}, b_{1}, b_{2}) =: (c_{0}, c_{1}, c_{2}).

Then there is a permutation π such that c_{π0} ≥ c_{π1} ≥ c_{π2}. We use again the projection

π(b_{0}, b_{1}, b_{2}) := (\frac{b_{1}}{b_{0}} , \frac{b_{2}}{b_{0}} ).

Then we arrive at the following map on B^{2} := {(x_{1}, x_{2}) : 0 ≤ x_{2} ≤ x_{1} ≤ 1}.

S(x_{1}, x_{2}) = (\frac{x_{1}}{1 − x_{2}} , \frac{x_{2}}{1 − x_{2}} ), 0 < x_{1} < 1, x_{1} + x_{2} < 1

S(x_{1}, x_{2}) = (\frac{1 − x_{2}}{x_{1}} , \frac{x_{2}}{x_{1}} ), 1 ≤ x_{1} + x_{2}, 0 ≤ x_{2} ≤ \frac{1}{2}

S(x_{1}, x_{2}) = (\frac{x_{2}}{x_{1}} , \frac{1 − x_{2}}{x_{1}} ), 1 ≤ x_{1} + x_{2}, \frac{1}{2} ≤ x_{2} ≤ 1.

A remarkable property about the Selmer algorithm is the fact that for almost all w ∈ B^{2} there is an n = n(w) such that for all m ≥ n, S^{m}w ∈ {(x_{1}, x_{2}) ∈ B^{2} : 1 ≤ x_{1} + x_{2}} (see [5]).

It is natural to consider the multiplicative acceleration of the algorithm, i.e., we perform in the first step

τ(b_{0}, b_{1}, b_{2}) := (b_{0} − kb_{2}, b_{1}, b_{2}), k : [\frac{b_{0}}{b_{1}} ], k ≥ 1.

The second step is to reorder again. In a similar way as before this leads to the map

T (x_{1}, x_{2}) = (\frac{x_{2}}{x_{1}} , \frac{1 − kx_{2}}{x_{1}} ), k(x) := [\frac{1}{x_{2}} ].

The time-1-partition has the cells

B(k) := \{x ∈ B^{2} : k(x) = k\} = \{x ∈ B^{2} : \frac{1}{k + 1} < x_{2} ≤ \frac{1}{k}\}.

Then we see that

T B(k) = {x ∈ B^{2} : kx_{1} + x_{2} ≥ 1}.

Therefore no cylinder is full and T B(k) is not a union of cylinders of rank 1.

2. Periodic Expansions for Multiplicative Selmer Algorithm

Let k_{1}, k_{2}, ... be a sequence of natural numbers. Then we define

B_{0}^{(−2)} = B_{1}^{(−2)} := 0, B_{2}^{(−2)} := 1
B_{0}^{(−1)} := 1, B_{1}^{(−1)} = B_{2}^{(−1)} := 0
B_{0}^{(0)} = B_{2}^{(0)} := 0, B_{1}^{(0)} := 1

and by recursion

B_{i}^{(s+1)} := k_{s+1}B_{i}^{(s−1)} + B_{i}^{(s−2)}, i = 0, 1, 2, s ≥ 0.

If we introduce matrices

β(k) := ( \begin{matrix}0 & k & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )

then we see that the relations

β^{(s)}(k_{1}, ..., k_{s}) := β(k_{1})...β(k_{s}) = ( \begin{matrix}B_{0}^{(s−1)} & B_{0}^{(s)} & B_{0}^{(s−2)} \\ B_{1}^{(s−1)} & B_{1}^{(s)} & B_{1}^{(s−2)} \\ B_{2}^{(s−1)} & B_{2}^{(s)} & B_{2}^{(s−2)} \end{matrix} )

hold. If y = T^{s}x and k_{i} = k(T^{i−1}x), 1 ≤ i ≤ s, then we find that

x_{i} = \frac{B_{i}^{(s−1)} + y_{1}B_{i}^{(s)} + y_{2}B_{i}^{(s−2)}}{B_{0}^{(s−1)} + y_{1}B_{0}^{(s)} + y_{2}B_{0}^{(s−2)}} , i = 1, 2.

Theorem 1 Assume that the algorithm of x = (x_{1}, x_{2}) eventually becomes periodic with period length p. Then x_{1} and x_{2} are rational functions in ρ, where ρ denotes the largest eigenvalue of the characteristic polynomial of the periodicity matrix β^{(p)}. Therefore x_{1} and x_{2} belong to a number field of degree ≤ 3.

Proof. Clearly, we can assume that the expansion is purely periodic with period length p. Let

Γ := β^{(p)}(k_{1}, ..., k_{p}) =: ( \begin{matrix}B_{0}^{(p−1)} & B_{0}^{(p)} & B_{0}^{(p−2)} \\ B_{1}^{(p−1)} & B_{1}^{(p)} & B_{1}^{(p−2)} \\ B_{2}^{(p−1)} & B_{2}^{(p)} & B_{2}^{(p−2)} \end{matrix} ) .

Note that

Γ^{k} = ( \begin{matrix}B_{0}^{(kp−1)} & B_{0}^{(kp)} & B_{0}^{(kp−2)} \\ B_{1}^{(kp−1)} & B_{1}^{(kp)} & B_{1}^{(kp−2)} \\ B_{2}^{(kp−1)} & B_{2}^{(kp)} & B_{2}^{(kp−2)} \end{matrix} ) .

Then the characteristic polynomial of Γ is given as

χ_{Γ}(t) := det(t1 − Γ) = t^{3} − At^{2} + Bt − 1.

If α, β, ρ are the eigenvalues we order them in a way such that |α| ≤ |β| < ρ. The Frobenius-Perron theorem says that 1 < ρ. Therefore min(|α|, |β|) < 1. Since αβρ = 1 the case α = β cannot occur. If the polynomial χ_{Γ}(t) has a multiple root then it would be reducible. Therefore if χ_{Γ}(t) = (x−β)^{2}(x−ρ) then ρ would be a rational number which satisfies |ρ| = 1, a contradiction. □

The difficulty for the present algorithm has two reasons. There is no convergence result known and max(B_{0}^{(s)}, B_{0}^{(s)−1}) can be B_{0}^{(s)} or B_{0}^{(s−1)}.

We first note that

B_{0}^{(p−1)} + B_{1}^{(p)} + B_{2}^{(p−2)} = A = ρ + β + α

and

ρβα = 1.

The Cayley-Hamilton Theorem (which was proved also by Perron in [4]) implies that

Γ^{3} − AΓ^{2} + BΓ − 1 = 0.

We multiply this equation with Γ^{k}β(k_{1}, ..., k_{j}) and see that for the entries of the matrices the relations

B_{i}^{((3+k)p+j)} − AB_{i}^{((2+k)p+j)} + BB_{i}^{((1+k)p+j)} − B_{i}^{(kp+j)} = 0, 0 ≤ i ≤ 2, 0 ≤ j < p

hold. The difference equation

a((3 + k)p) − Aa((2 + k)p) + Ba((1 + k)p) − a(kp) = 0

has the general solution

a(kp) = dρ^{k} + bβ^{k} + aα^{k}.

Remember that the case α = β does not occur in our situation. The numbers a, b, d are uniquely defined by the initial conditions

a(0) = d + b + a
a(p) = dρ + bβ + aα
a(2p) = dρ^{2} + bβ^{2} + aα^{2}.

Therefore we find

B_{i}^{(kp+j)} = d(i, j)ρ^{k} + b(i, j)β^{k} + a(i, j)α^{k}.

Note further that from the recursion relations

B_{i}^{(s+2)} = k_{s+2}B_{i}^{(s)} + B_{i}^{(s−1)}
B_{i}^{(s+3)} = k_{s+3}B_{i}^{(s+1)} + B_{i}^{(s)}

follows that if B_{i}^{(s)} ≫ ρ^{k} then B_{i}^{(s+j)} ≫ ρ^{k}, j ≥ 2. Therefore, if d(i, j) ≠ 0 for some j then d(i, j) ≠ 0 for all j, 0 ≤ j < p.

Now we know that

B_{0}^{(kp−1)} + B_{1}^{(pk)} + B_{2}^{(kp−2)} = ρ^{k} + β^{k} + α^{k}.

As noted before if y = T^{s}x then we find that

x_{i} = \frac{B_{i}^{(s−1)} + y_{1}B_{i}^{(s)} + y_{2}B_{i}^{(s−2)}}{B_{0}^{(s−1)} + y_{1}B_{0}^{(s)} + y_{2}B_{0}^{(s−2)}} , i = 1, 2.

But y = T^{s}x ∈ \{x ∈ B^{2} : \frac{1}{k_{s+1}+1} < x_{2} ≤ \frac{1}{k_{s+1}} \}.

Therefore

\frac{1}{k_{s+1} + 1} ≤ \frac{B_{i}^{(s−1)} + B_{i}^{(s)} + B_{i}^{(s−2)}}{B_{0}^{(s−1)} + B_{0}^{(s)} + B_{0}^{(s−2)}} ≤ k_{s+1} + 1, i = 1, 2.

If we take M = max(k_{1}, ..., k_{p}) + 1 then we obtain

\frac{1}{M} ≤ \frac{B_{i}^{(s−1)} + B_{i}^{(s)} + B_{i}^{(s−2)}}{B_{0}^{(s−1)} + B_{0}^{(s)} + B_{0}^{(s−2)}} ≤ M, i = 1, 2.

This implies that d(i, j) ≠ 0 for all i = 1, 2 and j =, 1, ..., p − 1.

From this we imply that

lim_{k→∞} \frac{B_{1}^{(kp)}}{B_{0}^{(kp)}} = \frac{d(1, 0)}{d(0, 0)} =: z_{1}, lim_{k→∞} \frac{B_{2}^{(kp)}}{B_{0}^{(kp)}} = \frac{d(2, 0)}{d(0, 0)} =: z_{2}

exist. The equation

Γ^{k} ( \begin{matrix}1 \\ x_{1} \\ x_{2} \end{matrix} ) = λ^{k} ( \begin{matrix}1 \\ x_{1} \\ x_{2} \end{matrix} )

shows that

B_{0}^{(kp−1)} + x_{1}B_{0}^{(pk)} + x_{2}B_{0}^{(kp−2)} = λ^{k}

for some positive eigenvalue of the periodicity matrix Γ. Since B_{0}^{(kp−1)} ∼ ρ^{k} we see that λ = ρ. Observe that z = (z_{1}, z_{2}) is a periodic point with the same expansion. Hence z = x. We can calculate x_{1} and x_{2} as rational functions in ρ from the equation

Γ ( \begin{matrix}1 \\ x_{1} \\ x_{2} \end{matrix} ) = ρ ( \begin{matrix}1 \\ x_{1} \\ x_{2}. \end{matrix} )

Remark 1 It is well known that a similar theorem holds true for the Jacobi-Perron algorithm ([4]). But there are some important differences.

(1) The characteristic polynomial is irreducible for the Jacobi-Perron algorithm but in our case the characteristic polynomial can be reducible. Let p = 1 and k_{1} = k. Then χ_{Γ}(t) = t^{3} − kt − 1. For k = 2 we see that β = −1. Note further that for k ≥ 3 we find β < −1. Here (x_{1}, x_{2}) = (\frac{1}{ρ}, \frac{1}{ρ^{2}} ).

(2) The Jacobi-Perron algorithm has periodic expansions of every (primitive) period p ≥ 1. For the multiplicative Selmer algorithm period p = 2 does not occur. Suppose that x = (x_{1}, x_{2}) has period 2. In this case we get x = (\frac{1}{λ(λ−k_{2})}, \frac{1}{λ}) for some eigenvalue λ. Therefore we get T x = (\frac{1}{λ(λ−k_{1})}, \frac{1}{λ}). Then k_{1} = [λ] = k_{2}. However, period p = 3 can occur. Let k_{1} = 2, k_{2} = 3, k_{3} = 5. Then we obtain ρ = 6.92167. The condition 0 ≤ x_{2} ≤ x_{1} ≤ 1 is satisfied since k_{1} + 1 < ρ, k_{2} + 1 < ρ, k_{3} + 1 < ρ.

Theorem 2 For periodic algorithms with α ≠ β we obtain the estimates

|B_{0}^{(pg)}x_{i} − B_{i}^{(pg)}| ≪ |β|^{g}, i = 1, 2

but there is at least one index i such that

max(|B_{0}^{(pg)}x_{i} − B_{i}^{(pg)}|, |B_{0}^{(pg+2)}x_{i} − B_{i}^{(pg+2)}|) ≫ |β|^{g}

holds for infinitely many values of g.

Proof. Since the quantities B_{0}^{(pg)}x_{i} − B_{i}^{(pg)} obey the same recursion relation we find

B_{0}^{(pg)}x_{i} − B_{i}^{(pg)} = k(i, 0)ρ^{g} + m(i, 0)β^{g} + n(i, 0)α^{g}, i = 1, 2.

As we have just shown

lim_{k→∞} \frac{B_{1}^{(kp)}}{B_{0}^{(kp)}} = x_{1}, lim_{k→∞} \frac{B_{2}^{(kp)}}{B_{0}^{(kp)}} = x_{2}

and since B_{0}^{(pg)} ∼ ρ^{g} we get k(1, 0) = k(2, 0) = 0. Hence for j = 1 and j = 2 we find

B_{0}^{(pg)}x_{i} − B_{i}^{(pg)} = m(i, 0)β^{g} + n(i, 0)α^{g}, i = 1, 2.

This shows that the first part of the theorem is true.

Now we will prove the second part of Theorem 2. In a similar way as before we find

B_{0}^{(pg+2)}x_{i} − B_{i}^{(pg+2)} = m(i, 2)β^{g} + n(i, 2)α^{g}, i = 1, 2.

It is easy to verify that (see Schweiger 2000, p.5)

det ( \begin{matrix}B_{0}^{(pg+2)}x_{1} − B_{1}^{(pg+2)} & B_{0}^{(pg)}x_{1} − B_{1}^{(pg)} \\ B_{0}^{(pg+2)}x_{2} − B_{2}^{(pg+2)} & B_{0}^{(pg)}x_{2} − B_{2}^{(pg)} \end{matrix} ) = \frac{1}{B_{0}^{(pg+1)} + B_{0}^{(pg+2)}y_{1} + B_{0}^{(pg)}y_{2}} .

Here T^{p}(x_{1}, x_{2}) = (y_{1}, y_{2}). Note that from B_{0}^{(gp)} ∼ ρ^{g} we obtain

max_{i=1,2}(|B_{0}^{(pg)}x_{i} − B_{i}^{(pg)}|, |B_{0}^{(pg+2)}x_{i} − B_{i}^{(pg+2)}|) ≫ \frac{1}{(\sqrt{ρ})^{g}} = \sqrt{|αβ|^{g}}.

For the last equality we used αβρ = 1. If |α| = |β| then

max_{i=1,2}(|B_{0}^{(pg)}x_{i} − B_{i}^{(pg)}|, |B_{0}^{(pg+2)}x_{i} − B_{i}^{(pg+2)}|) ≫ |β|^{g}.

If |α| < |β| then at least one of the numbers m(1, 0), m(2, 0), m(1, 2), m(2, 2) cannot vanish. If m(1, 0) = m(2, 0) = m(1, 2) = m(2, 2) = 0 then we see that

ρ^{−g} ≪ det ( \begin{matrix}B_{0}^{(pg+2)}x_{1} − B_{1}^{(pg+2)} & B_{0}^{(pg)}x_{1} − B_{1}^{(pg)} \\ B_{0}^{(pg+2)}x_{2} − B_{2}^{(pg+2)} & B_{0}^{(pg)}x_{2} − B_{2}^{(pg)} \end{matrix} ) ≪ |α|^{2g}

which (by using αβρ = 1) leads to |β| ≤ |α|. Therefore we get again

max_{i=1,2}(|B_{0}^{(pg)}x_{i} − B_{i}^{(pg)}|, |B_{0}^{(pg+2)}x_{i} − B_{i}^{(pg+2)}|) ≫ |β|^{g}.

□

Remark 2 Since |β| > 1 can occur we see that lim sup_{g→∞} |B_{0}^{(pg)}x_{i} − B_{i}^{(pg)}| = ∞ is possible. Therefore a generalization of Nakaishi’s approach [3] seems to be out of reach.

3. Periodic Expansions for the Triangle Sequence

The shape of the time-1-matrices for Selmer’s multiplicative algorithm suggest to investigate the algorithm with time-1-matrices

β(k) := ( \begin{matrix}1 & k & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) .

These matrices belong to an algorithm which has been called the triangle sequence by Garrity [2] (see also [1]). The underlying map is given on B^{2} by

T (x_{1}, x_{2}) = (\frac{x_{2}}{x_{1}} , \frac{1 − x_{1} − kx_{2}}{x_{1}} ), k = [\frac{1 − x_{1}}{x_{2}} ].

There are some similarities but also some differences between these algorithms. The most important difference is the time-1-partition. The cylinders B(k) are the triangles with vertices (1, 0), (\frac{1}{k+1}, \frac{1}{k+1}), (\frac{1}{k+2}, \frac{1}{k+2}), k ≥ 0. Therefore all cylinders are full, i. e. T B(k) = B^{2}, k ≥ 0. If we set

β^{(s)}(k_{1}, ..., k_{s}) := β(k_{1})...β(k_{s}) =: ( \begin{matrix}B_{00}^{(s)} & B_{01}^{(s)} & B_{00}^{(s−1)} \\ B_{10}^{(s)} & B_{11}^{(s)} & B_{10}^{(s−1)} \\ B_{20}^{(s)} & B_{21}^{(s)} & B_{20}^{(s−1)} \end{matrix} )

we find the recursion relations

B_{i0}^{(s+1)} = B_{i0}^{(s)} + B_{i1}^{(s)}, i = 0, 1, 2
B_{i1}^{(s+1)} = k_{s+1}B_{i0}^{(s)} + B_{i0}^{(s−1)}, i = 0, 1, 2.

Similar to the multiplicative Selmer algorithm we see easily the following properties. As before let α, β, ρ be the three eigenvalues of the characteristic polynomial ordered in a way such that |α| ≤ |β| < ρ.

(1) The characteristic polynomial can be reducible. Let p = 1 and k_{1} = k. Then χ_{Γ}(t) = t^{3} − t^{2} − kt − 1. For k = 3 we see that β = −1.

(2) It is possible to have 0 < α < 1 < β < ρ. Let p = 2 and k_{1} = 4, k_{2} = 3. Then χ_{Γ}(t) = t^{3} − 5t^{2} + 6t − 1. Then ρ = 6.480786620527056, β = 1.4097605688346069, α = 0.109452810638337.

In contrast to the multiplicative Selmer algorithm much more is known about convergence (see [1]). In fact, it is known that the triangle sequence is convergent for periodic expansions. This makes it easier to prove the following theorem, which is a strengthening of Theorem 9 of [2].

Theorem 3 Assume that the algorithm of x = (x_{1}, x_{2}) eventually becomes periodic with period length p. Then x_{1} and x_{2} are rational functions in ρ, where ρ denotes the greatest eigenvalue of the characteristic polynomial of the periodicity matrix β^{(p)}. Therefore x_{1} and x_{2} belong to a number field of degree ≤ 3.

Proof. Since the proof closely follows Perron’s proof ([4],[5]), we will only sketch it. Let (x_{1}, x_{2}) have a purely periodic expansion with periodicity matrix

Γ = β^{(p)} = ( \begin{matrix}B_{00}^{(p)} & B_{01}^{(p)} & B_{00}^{(p−1)} \\ B_{10}^{(p)} & B_{11}^{(p)} & B_{10}^{(p−1)} \\ B_{20}^{(p)} & B_{21}^{(p)} & B_{20}^{(p−1)} \end{matrix} ) .

Then

( \begin{matrix}B_{00}^{(p)} & B_{01}^{(p)} & B_{00}^{(p−1)} \\ B_{10}^{(p)} & B_{11}^{(p)} & B_{10}^{(p−1)} \\ B_{20}^{(p)} & B_{21}^{(p)} & B_{20}^{(p−1)} \end{matrix} ) ( \begin{matrix}1 \\ x_{1} \\ x_{2} \end{matrix} ) = λ ( \begin{matrix}1 \\ x_{1} \\ x_{2} \end{matrix} )

for some eigenvalue λ > 1. Clearly, x_{1} and x_{2} are rational functions in λ. From Γ^{p+1} = Γ^{p}Γ we obtain the relation

\frac{B_{00}^{((g+1)p)}}{B_{00}^{(gp)}} = B_{00}^{(p)} + B_{01}^{(p)} \frac{B_{10}^{(gp)}}{B_{00}^{(gp)}} + B_{00}^{(p−1)} \frac{B_{20}^{(gp)}}{B_{00}^{(gp)}} .

From the convergence of the algorithm we obtain

lim_{g→∞} \frac{B_{00}^{((g+1)p)}}{B_{00}^{(gp)}} = B_{00}^{(p)} + B_{01}^{(p)}x_{1} + B_{00}^{(p−1)}x_{2} = λ.

Using this result and the recursion relations we obtain B_{00}^{(gp)} ∼ λ^{g} and B_{01}^{(gp)} ∼ λ^{g}. However from

B_{00}^{(gp)} + B_{11}^{(gp)} + B_{20}^{(gp−1)} = ρ^{g} + α^{g} + β^{g}

we see that λ = ρ. □

Remark 3 Clearly Theorem 2 can also be extended to the triangle sequence.

Acknowledgments. I am indebted to B. Schratzberger and the referee for valuable comments and suggestions. This research was partially supported by the Austrian Science Foundation FWF, Project Number P16964-N12.

References

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