Journal de Théorie des Nombres de Bordeaux

Aminata Keita

Aminata KEITA
Continued fractions and Parametric geometry of numbers
Tome 29, no 1 (2017), p. 129-135.
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Journal de Théorie des Nombres
de Bordeaux 29 (2017), 129–135
Continued fractions
and
Parametric geometry of numbers
par Aminata KEITA
Résumé. W.M. Schmidt et L. Summerer ont récemment proposé
une théorie qui décrit approximativement le comportement des
minima successifs d’une famille de corps convexes à un paramètre,
celle qui intervient dans les problèmes d’approximation rationnelle
simultanée de plusieurs nombres. Dans le cas d’un seul nombre,
nous montrons que le comportement qualitatif exact des minima
reflète le développement en fraction continue de la plus petite
distance de ce nombre à un entier.
Abstract. Recently, W. M. Schmidt and L. Summerer devel-
oped a new theory called Parametric Geometry of Numbers which
approximates the behaviour of the successive minima of a family
of convex bodies in Rn related to the problem of simultaneous
rational approximation to given real numbers. In the case of one
number, we show that the qualitative behaviour of the minima
reflects the continued fraction expansion of the smallest distance
from this number to an integer.
1. Introduction
The parametric geometry of numbers, recently introduced by W. M.
Schmidt and L. Summerer, is a theory which analyzes the behaviour of
the successive minima of parametric families of convex bodies in Rn. It
provides a new approach to the problem of simultaneous approximation to
real numbers by rational numbers. It was initially developed in dimension
n = 3 in [3], then extended to the general case n > 2 in [4], and completed
in [1]. Our goal is to revisit the case n = 2 by providing an exact description
of the qualitative behaviour of the successive minima in that case. For this,
we consider the family of convex bodies given by
Cξ(eq) := {(x, y) ∈ R2 ; |x| 6 eq , |xξ − y| 6 e−q} (q > 0),
Manuscrit reçu le 30 janvier 2015, révisé le 13 mars 2015, accepté le 15 mars 2016.
Mathematics Subject Classification. 11J04, 11J82.
Mots-clefs. Convex body, Successive minima, Combined graph, Continue fractions.
Work partially supported by NSERC (Canada).
130 Aminata Keita
for some fixed ξ ∈ R. For j = 1, 2 and q > 0, let Lξ,j(q) = log λj(Cξ(eq)),
where λj(Cξ(eq)) denotes the j-th minimum of Cξ(eq), i.e. the smallest λ > 0
such that λCξ(eq) contains at least j linearly independent elements in Z2.
Further, we define a function Lξ : [0,∞) −→ R2 by
Lξ(q) = (Lξ,1(q), Lξ,2(q)) (q > 0).
As noted by Schmidt and Summerer [3, §4], Lξ enjoys the following prop-
erties.
• Each component Lξ,j : [0,∞) → R is continuous and piecewise
linear with slope ±1.
• If q ∈ [0,∞) is such that Lξ,1 admits a local maximum at q, then
we have
Lξ,1(q) = Lξ,2(q).
• For all q > 0, we have − log 2 6 Lξ,1(q) + Lξ,2(q) 6 0 (by virtue of
Minkowski’s theorem.)
The union of the graphs of Lξ,1 and Lξ,2 in [0,∞)×R is called the combined
graph of the function Lξ, and is denoted Γξ. Our objective is to show that
the simple continued fraction expansion of a number ξ ∈ R can be read
from this graph when ξ ∈ [0, 12 ].
Theorem 1.1. Let ξ ∈ R. Then, Γξ = Γ‖ξ‖ depends only on the distance
from ξ to the closest integer, denoted ‖ξ‖. Moreover, let (qn)06n<s with
s ∈ N? ∪ {∞} denote the increasing sequence of points in [0,∞) for which
Lξ,1 admits a local maximum, and let an denote for each positive integer
n < s the number of local maxima of Lξ,2 restricted to the interval [qn−1, qn].
Then it follows that q0 = 0, and that the simple continued fraction expansion
of ‖ξ‖ is given by
‖ξ‖ =

[0] if s = 1,
[0, a1, a2, . . . , as−1] if 2 6 s <∞,
[0, a1, a2, . . .] if s =∞,
(1.1)
with as−1 > 2 if 2 6 s <∞.
In particular, the numbers an are the partial quotients of ‖ξ‖. Figure 1.1
illustrates this result by showing the combined graph Γξ (in solid lines) for
several rational numbers ξ.
For the proof, fix an arbitrary ξ ∈ R, and choose m ∈ Z and ε = ±1 such
that ξ = m+ ε‖ξ‖. The map
φ : R2 −→ R2
(x, y) 7−→ (x, ε(y −mx))
Parametric geometry of numbers 131
qq0
a) 0 = [0]
qq0
q1
b) 1/2 = [0, 2]
qq0
q1 q2
c) 3/7 = [0, 2, 3]
Figure 1.1. Examples of combined graphs.
is an R-vector space isomorphism which satisfies φ(Z2) = Z2. Thus, for
each q > 0, we have that
λj(Cξ(eq)) = λj(φ(Cξ(eq)) = λj(C‖ξ‖(eq)) (j = 1, 2),
from which it follows that Lξ(q) = L‖ξ‖(q). This proves the first statement
of the theorem.
For what follows, we shall suppose that ξ = ‖ξ‖ ∈ [0, 12 ]. The simple
continued fraction expansion of ξ ∈ [0, 12 ] is given by (1.1) for some sequence
(an)16n<s in N?, with as−1 > 2 if 2 6 s < ∞. The n-th convergent of ξ is
the rational number
Pn
Qn
= [0, a1, . . . , an] (1 6 n < s),
where Pn and Qn are the positive integers given by the recurrence formula
Qn = Qn−2 + anQn−1 and Pn = Pn−2 + anPn−1 (1 6 n < s),
with initial values P−1 = Q0 = 1 and P0 = Q−1 = 0. Let Qs =∞ if s <∞.
The proof of the second part of the theorem uses the following facts (see [2,
chap I]).
(i) The sequence (Qn)06n<s is a strictly increasing sequence of positive
integers.
(ii) The sequence (Qnξ − Pn)−16n<s consists of real numbers of alter-
nating signs (except the last term which is zero if s < ∞), whose
absolute values are strictly decreasing.
(iii) We have QnPn−1 − Qn−1Pn = (−1)n for each integer n satisfying
0 6 n < s.
Consider the integers
(1.2) Qn,t = Qn−2 + tQn−1, Pn,t = Pn−2 + tPn−1 (0 6 t 6 an, 1 6 n < s).
132 Aminata Keita
The fractions Pn,t/Qn,t with 0 < t < an, when they exist, are called the
semiconvergents of ξ between Pn−1/Qn−1 and Pn/Qn. The following propo-
sition relates the points xn,t = (Qn,t, Pn,t) to the points xn = (Qn, Pn) with
−1 6 n < s, as well as to the quantities ∆n and ∆n,t defined by
∆n = |Qnξ − Pn| (−1 6 n < s)
and
∆n,t = |Qn,tξ − Pn,t| (0 6 t 6 an , 1 6 n < s).
Proposition 1.2. Let n be a positive integer with n < s. Then we have
Qn,0 = Qn−2 < Qn−1 6 Qn,1 < · · · < Qn,an = Qn,(1.3)
∆n,an = ∆n < ∆n−1 6 ∆n,an−1 < · · · < ∆n,0 = ∆n−2,(1.4)
with Qn,1 = Qn−1 iff n = 1, and ∆n−1 = ∆n,an−1 iff s <∞ and n = s− 1.
Moreover, the points xn,t = (Qn,t, Pn,t) with 0 6 t 6 an are precisely the
pairs of non-negative integers (Q,P ) satisfying
(1.5)
Qn−2 6 Q 6 Qn, |Qξ − P | 6 ∆n−2 = |Qn−2ξ − Pn−2|,
and QPn−1 −Qn−1P 6= 0.
Finally, if s <∞, then there exists no pair (Q,P ) ∈ Z2\{0} satisfying
(1.6) |Qξ − P | < ∆s−2 = |Qs−2ξ − Ps−2|, QPs−1 −Qs−1P 6= 0.
Due to the lack of a convenient reference, we give a succinct proof.
Proof. Since Qn−2ξ − Pn−2 and Qn−1ξ − Pn−1 are non-zero, of opposing
signs, and since
Qnξ − Pn = (Qn−2ξ − Pn−2) + an(Qn−1ξ − Pn−1)
is either zero or of the same sign as Qn−2ξ − Pn−2, it follows that
∆n,t = ∆n−2 − t∆n−1 (0 6 t 6 an),
and that ∆n,an−1 = ∆n,an + ∆n−1 = ∆n + ∆n−1. The first assertion of the
Proposition then follows from properties (i) and (ii).
Let x = (Q,P ) be a non-zero integer point satisfying (1.5). Since the
points xn−2 = (Qn−2, Pn−2) and xn−1 = (Qn−1, Pn−1) form a basis of Z2
(by virtue of (iii)), we may write
x = rxn−2 + txn−1,
with (r, t) ∈ Z2\{0}. The third condition of (1.5) yields r 6= 0. Thus, the
equality
Qξ − P = r(Qn−2ξ − Pn−2) + t(Qn−1ξ − Pn−1)
combined with property (ii) and the second condition of (1.5) implies that
rt > 0 and that t > an if r > 1. Since Q = rQn−2 + tQn−1, then (i) allows
us to conclude that r = 1 and that 0 6 t 6 an.
Parametric geometry of numbers 133
Finally, if s <∞ and x = (Q,P ) satisfies (1.6), we may write
x = axs−2 + bxs−1 with a, b ∈ Z and a 6= 0.
Since ∆s−1 = 0, we deduce that |Qξ − P | = |a|∆s−2 > ∆s−2, contrary to
the hypothesis. 
For each x = (Q,P ) ∈ Z2 and each q > 0, we define Lx(q) to be the
logarithm of the smallest real number λ > 0 such that x ∈ λCξ(eq). A
simple computation gives
Lx(q) = max{log |Q| − q , log |Qξ − P |+ q} (q > 0).
The graph of the function Lx : [0,∞) → R is called the trajectory of
the point x. The union of these graphs for x ∈ Z2\{0} primitive (i.e. with
coprime coordinates) contains Γξ (see [3, §4] or [1, §2]). In fact the following
smaller set of points suffices.
Proposition 1.3. The graph Γξ is covered by the trajectories of the points
xn with −1 6 n < s and those of the points xn,t with 1 6 n < s and
0 < t < an.
Proof. Indeed, let x = (Q,P ) ∈ Z2 be primitive with ±x different from all
these points. Then we have Q 6= 0 otherwise x = (0,±1) = ±x−1 against
the hypothesis. Since Lx = L−x, we may assume without loss of generality
that Q > 1. Thus, there exists an integer n with 1 6 n 6 s such that
Qn−1 6 Q < Qn, while Proposition 1.2 yields |Qξ−P | > ∆n−2. We deduce
from (1.3) and (1.4) that
Lx(q) > max {logQn−1 − q, log ∆n−2 + q} = max{Lxn−2(q), Lxn−1(q)},
for all q > 0. Thus, since xn−2 and xn−1 are linearly independent, the
trajectory of x does not contribute to the cover. 
The inequalities of (1.3) and (1.4) of Proposition 1.2 yield
Lxn,t(q) > Lxn−1(q) (q > 0, 1 6 n < s, 0 < t < an).(1.7)
Since Lx−1(q) = q > Lx0(q) for all q > 0, it follows that the graph of Lξ,1 is
covered by the trajectories of the points xn with 0 6 n < s, and thus that
Lξ,1(q) = min {Lxn(q) ; 0 6 n < s} (q > 0).
In particular, we have Lξ,1(q) = Lx0(q) = −q near q = 0. Thus, q0 = 0 is
the first local maximum of Lξ,1 on [0,∞). Let n be a positive integer with
n < s. The inequalities of (1.3) and (1.4) also imply that the trajectories
of xn−1 and of xn meet in a single point. To the left of this point, the
trajectory of xn−1 has slope +1, while to the right, the trajectory of xn has
134 Aminata Keita
slope −1. The abscissa of this point is thus the (n+ 1)-th local maximum
qn of Lξ,1 on [0,∞). It is given by
qn =
1
2 (logQn − log ∆n−1) .
Moreover, Lξ,1 coincides with Lxn−1 on the interval [qn−1, qn] (see Fig-
ure 1.2). Since none of the points xn,0 = xn−2, xn,1, . . . ,xn,an = xn is a
multiple of xn−1, we further deduce from (1.7) that, on the same interval,
we have
Lξ,2(q) = min
{
Lxn,t(q) ; 0 6 t 6 an
}
(qn−1 6 q 6 qn).
If n 6= 1 and n 6= s− 1, each inequality of (1.3) and of (1.4) is strict. Thus,
the trajectories of the points xn,t with 0 6 t 6 an are a part of the graph
of Lξ,2 over [qn−1, qn], as shown in Figure 1.2.b. The points at which the
trajectories of xn,t−1 and of xn,t cross for t = 1, . . . , an give the an local
maxima of Lξ,2 on [qn−1, qn], each contained in the interior of this interval.
If n = 1 and s > 2, the situation is illustrated by Figure 1.2.a. In this
case, q0 = 0 is the first local maximum of Lξ,2 on [q0, q1]. If s < ∞ and if
n = s− 1 > 1, then qs−1 is the last local maximum of Lξ,2 on [qs−2, qs−1],
as illustrated in Figure 1.2.c. Finally, if s = 1, we have that ξ = 0 and
the graph Γ0 is given by Figure 1.1.a. In general, if s < ∞, we have that
Lξ,2(q) = Lxs−2(q) = q + log ∆s−2 for all q > qs−1, from which it follows
that Lξ,2 admits no local maximum on [qs−1,∞).
q0
x−1
x0
x1,1
x1,3
x1,2
a) to the right of q0
qn−1 qn
xn−2
xn
xn−1
xn,an−1
· · ·
xn,2
xn,1
b) over [qn−1, qn]
if 2 6 n < s− 1
qs−1
xs−1
xs−1,as−1−1
xs−1,as−1−2
xs−2
c) near qs−1
if 2 6 s <∞
Figure 1.2. Combined graph Γξ.
Parametric geometry of numbers 135
References
[1] D. Roy, “On Schmidt and Summerer parametric geometry of numbers”, Annals of Mathe-
matics, to appear.
[2] W. M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785,
Springer-Verlag, 1980, x+299 pages.
[3] W. M. Schmidt & L. Summerer, “Parametric geometry of numbers and applications”, Acta
Arith. 140 (2009), no. 1, p. 67-91.
[4] ———, “Diophantine approximation and parametric geometry of numbers”, Monatsh. Math.
169 (2013), no. 1, p. 51-104.
Aminata Keita
Département de Mathématiques
Université d’Ottawa
585 King Edward
Ottawa, Ontario K1N 6N5, Canada
E-mail: akeit104@uottawa.ca


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