Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

60 Pages·2010·0.73 MB·English

Quantitative Density under Higher Rank Abelian Algebraic Toral
Actions
Zhiren Wang∗
Abstract
We generalize Bourgain-Lindenstrauss-Michel-Venkatesh’s recent one-dimensional quantita-
tive density result to abelian algebraic actions on higher dimensional tori. Up to finite index,
the group actions that we study are conjugate to the action of U , the group of units of some
K
non-CM number field K, on a compact quotient of K ⊗QR. In such a setting, we investigate
how fast the orbit of a generic point can become dense in the torus. This effectivizes a spe-
cial case of a theorem of Berend; and is deduced from a parallel measure-theoretical statement
which effectivizes a special case of a result by Katok-Spatzier. In addition, we specify two
numerical invariants of the group action that determine the quantitative behavior, which have
number-theoretical significance.
∗Department of Mathematics, Princeton University, Princeton, NJ 08544, USA; [email protected]
Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Organization of paper and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Abelian algebraic actions on the torus 5
2.1 Preliminaries on number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Construction of the number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Maximal rank and the full group of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Total irreduciblity and non-CM number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Action of the group of units 11
3.1 Bounded totally irreducible units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Characters on X and irrationality of eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Positive entropy and regularity 14
4.1 Some non-conventional entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Positive entropy in an eigenspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 From entropy to L2-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Group action on a single eigenspace 23
5.1 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Approximation of an arithmetic progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 Escape from a fixed line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Measure-theoretical results 35
6.1 Fourier coefficients on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.2 Manipulation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 The effective measure-theoretical theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Topological results 44
7.1 Density of the orbit of a dispersed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.2 Density of the orbit of a large set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3 Density of the orbit of a single point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8 Appendix: A number-theoretical application 53
1 Introduction
1.1 Background
The rigidity of higher rank abelian algebraic actions has since long been studied in various
forms. The first result of this type was achieved by Furstenberg’s disjointness theory :
Theorem 1.1. (Furstenberg [8], ’67) Any minimal closed subset of T = R/Z simultaneously in-
variant under ×2 and ×3 is either T itself or a finite set of rational points.
Themeasure-theorecticalanalogueofthistheoremisthefamousFurstenberg’sconjecture,which
asks whether any ergodic ×2,×3-invariant measure on T is either uniform or finitely supported on
rational points. The conjecture remains open.
There are several ways to extend Furstenberg’s theorem. One of them is achieved by Berend,
who proved an analogue on higher dimensional tori.
Theorem 1.2. (Berend [2], ’83) Let Σ < M (Z) = End(Td) be an abelian subsemigroup such that:
d
(i) for any common eigenspace V of Σ, there is an element g whose eigenvalue corresponding to
V has absolute value strictly greater than 1; (ii) rank(Σ) ≥ 2; (iii) Σ contains a totally irreducible
element. Then any minimal Σ-invariant closed set on Td is either the full torus or finite.
HeretheactionofasingletoralendomorphismA ∈ M (Z)isirreducibleifthereisnonon-trivial
d
A-invariant subtorus on Td = Rd/Zd; it is totally irreducible if An is irreducible for every positive
integer n.
Another way of extension was to go to the measure-theoretical category. Under the assump-
tion of positive entropy, Furstenberg’s conjecture has been proved by Rudolph following work of
Lyons[17].
Theorem 1.3. (Rudolph[19],’90 Suppose a Borel probability measure µ on T is invariant and
ergodic under both ×2 and ×3. If the measure-theoretical entropy h (×2) of the ×2 action with
µ
respect to µ is positive, then µ is the Lebesgue measure.
ThisresultwasextendedlaterbyJohnsontothejointactionbyanymultiplicativelyindependent
pair of positive integers (which are not necessarily coprime) in [12].
Itispossibletopursuethesetwodirectionsofextensionatthesametime. Namely,undercertain
conditions, foranabelianactiononTd withsomekindofhyperbolicity, higherrankandirreducibil-
ity, any ergodic invariant measure with positive entropy is expected to be uniform. This was first
proved by Katok and Spatzier[14] for abelian subgroups of SL (Z) under a special assumption
d
called total non-symplecticity (TNS); for a detailed treatment of their result, see Kalinin-Katok[13,
§3]. Later on, such a measure rigidity statement was proved by Einsiedler-Lindenstrauss[6] for a
more general family of groups of toral automorphisms.
Further, there is a third way to extend these results: all the theorems mentioned above have
quantitative versions. Recently [3], Bourgain, Lindenstrauss, Michel and Venkatesh effectivized
Furstenberg’s theorem.
Theorem 1.4. (Bourgain-Lindenstrauss-Michel-Venkastesh[3],’08) For any pair of multiplicatively
independent positive integers a,b:
(i). If x ∈ T is diophantine generic:
p
|x− | ≥ q−k, ∀p,q ∈ Z, q ≥ 2,
q
1
then {ambnx|0 < m,n ≤ N} is (loglogQ)−c-dense in T for all Q ≥ Q , where c = c(a,b) and
0
Q = N (k,a,b) are constants.
0 0
(ii). Ifx = p whereQiscoprimewithab, then{ambnx|0 < m,n < 3logQ}isC(logloglogQ)−c-
Q
dense in T where C = C(a,b) and c is the same as in (i).
Roughly speaking, this says the orbit of a point x is of certain quantitative density unless x is
very close to a rational number with small denominator.
1.2 Statement of main results
In this paper we generalize Bourgain-Lindenstrauss-Michel-Venkatesh’s theorems to the higher
dimensional case. We investigate a special case of the situation studied by Berend [2], namely the
action on Td by a group of toral automorphisms G < SL (Z) that satisfies:
d
Condition 1.5. G is an abelian subgroup of SL (Z) such that:
d
(1). rank(G) ≥ 2, where rank refers to the torsion free rank of finitely generated abelian groups.;
(2). G contains an totally irreducible toral automorphism;
(3). G is maximal in rank: there is no intermediate abelian subgroup G in SL (Z) containing
1 d
G such that rank(G) < rank(G ) .
1
We now equips G with a norm.
Definition 1.6. (i). For a matrix A ∈ M (R), we define the logarithmic Mahler measure of
d
A to be
1 Z 2π
m(A) := log|det(A−eiθid)|dθ.
2π
0
An alternative definition is
d
X
m(A) = log |ζi |, (1.1)
+ A
i=1
where ζ1,··· ,ζd are the eigenvalues of A and log x = max(0,logx).
A A +
(ii). For a subgroup G < SL (Z) and L > 0, let BMah(L) be the ball of radius L with respect to
d G
logarithmic Mahler measure:
BMah(L) := {g ∈ G|m(g) ≤ L}.
G
For more information on Mahler measures, we refer to [7].
Here is our main result in the topological category:
Theorem 1.7. If an abelian subgroup G < SL (Z) satisfies Condition 1.5 then there are effective
d
constants c , c , c , and c depending only on G such that if a finite set E ⊂ Td is (cid:15)-separated
1 2 3 4
and |E| ≥ (cid:15)−αd for some α,(cid:15) > 0 with (cid:15) ≤ c , α ≥ c logloglog1(cid:15) then BMah(c log 1).E := {g.x|g ∈
1 2 loglog1 G 3 (cid:15)
(cid:15)
BGMah(c3log 1(cid:15)),x ∈ E} is (log 1(cid:15))−c4α-dense.
The proof of Theorem 1.7 is based on the analogous effective measure-theoretical Theorem 1.9
below which studies the behaviour of a measure µ under the G-action. We assume µ has positive
entropy up to a certain scale, which is formulated by the following condition:
2
Condition 1.8. µ is a Borel measure on Td such that for some given pair α > 0, (cid:15) > 0, the entropy
H (P) = P −µ(P)logµ(P) is at least αdlog 1 for all measurable partitions P of Td such that
µ P∈P (cid:15)
diamP ≤ (cid:15), where diamP is the maximal diameter of atoms from P.
Theorem 1.9. Suppose an abelian subgroup G < SL (Z) meets Condition 1.5. There are effective
d
constants c , c , c , and c depending only on G such that if (cid:15) ≤ c , δ ∈ [c (loglog 1)−1, α] and a
1 3 5 6 1 5 (cid:15) 2
Borel measure µ on Td satisfies the entropy condition 1.8 then there exists a measure µ0 which has
totalmass|µ0| ≥ α−δ andisdominatedbysomeelementµ00 fromtheconvexhullofBMah(c log 1).µ
G 3 (cid:15)
in the space of probability measures on Td (here the group G acts on µ by pushforward: g.µ = g µ),
∗
so that ∀f ∈ C∞(Td),
Z
(cid:12)(cid:12)µ0(f)−|µ0| Tdf(x)dx(cid:12)(cid:12) ≤ c6(log 1(cid:15))−12c5−1δkfkH˙ d+21.
Theorem 1.9 actually effectivizes a special case of Katok-Spatzier’s result [14].
AnotherinterestingcorollarytoTheorem1.7willbeTheorem1.10, regardinghowfastageneric
single G-orbit can fill up Td, which is a quantitative form of the fact that any infinite G-orbit is
dense (proved by Berend [2]) and the generalization of a similar theorem in [3].
Theorem 1.10. Suppose G satisfies Condition 1.5 then there are effective constants c , c depend-
7 8
ing only on G such that:
For all Q ∈ N, Q ≥ c , if a point x ∈ Td = Rd/Zd satisfies one of the following conditions:
7
(i) x is diophantine generic: ∃k > 1 such that |x− v| ≥ q−k for any coprime pair (v,q) where
q
q ∈ N and v ∈ Zd; OR
(ii) x = v where v ∈ Zd is coprime with Q, in which case we denote k = 1,
Q
then the set BMah(cid:0)(k+2)logQ(cid:1).x is (logloglogQ)−c8-dense.
G
Remark 1.11. In fact, in all the theorems stated above, we are able to know on which features
of G the constants really depend and how the dependence looks like. Actually, all constants are
determined by the dimension d and two algebraic invariants M and F of G. For the explicit
ψ φ(G)
expressions, see Propositions 6.10, 7.1, 7.2, 7.6.
The exact definitions of M and F are going to appear later in §2. However, essentially
ψ φ(G)
M is a measurement of how “twisted” the eigenbasis of G is (notice G is commutative so all its
ψ
elements share a common eigenbasis); and F is the bound on a set of generators of G (up to
φ(G)
finite-index) in terms of logarithmic Mahler measure.
We make an effort to track the dependence on M and F in this paper, the reason is that
ψ φ(G)
this dependence can be interesting from a number-theoretical point of view (cf. Appendix §8 for
example).
Remark 1.12. It should be remarked that if k·k is the word length metric with respect to some
WL
fixed generating set S ⊂ G, then m(g) . kgk + 1,∀g ∈ G. Therefore in all the theorems
G,S WL
stated above, the ball BMah(c log 1) with respect to Mahler measure can be replaced by some ball
G 3 (cid:15)
BWL(c 0log 1) defined in terms of word length metric, where c 0 = c 0(G,S) is effective.
G 3 (cid:15) 3 3
A major restricion we adopted in addition to Berend’s conditions is that G is supposed to be
maximal in rank, which guarantees that one can expand an arbitrary eigenspace while contracting
everythingelse. Thestudyofthecasewithoutthisassumptionisalsopossible, thoughmorecareful
argumentsarerequiredandonlyweakerestimateswillbeobtained,andishopefullygoingtoappear
in another paper [16].
3
1.3 Organization of paper and notations
While our proof follows the structure of that of [3] which uses Fourier analysis, we also borrow
several arguments from [2]. A little bit of number theory is needed as our group G turns out to
be closely linked to the full unit group of some number field. There are some new difficulties to
be treated in the higher dimensional case. The main point is that instead of the case on a one
dimensional circle, we are going to decompose a multidimensional torus into eigenspaces and work
on one of them. We will show some kind of “dense” distribution along a line in that eigenspace and
then make use of the irreduciblity of the action (so any eigenspace does not form a close subtorus).
One of the difficulties here is that when the eigenspace is complex (i.e. 2-dimensional), a line in it
may not be equidistributed in Td, this is going to be dealt with in §5.3
The organization of paper is as follows:
§2 discusses the number-theoretical implication of Condition 1.5 and proves the G-action comes
from the group of units of a certain number field and its eigenspaces correspond to the real and
complexembeddingsofthatfield. Further, in§3wediscussafewalgebraicpropertiesofthisaction,
in particular the irrationality of the eigenspaces is specified quantitatively.
In §4 we give an effective description of the fact that if a measure on Td has positive entropy
then its projection to at least one of the eigenspaces V has positive entropy. Since the restriction of
i
the group action on V is just a multiplication, we are able to discuss its behaviors in §5. Roughly
i
speaking, we are going to show that using the group action, it is possible to stretch a short vector,
to generate an approximation of a line segment (or an arithmetic progression) from a given vector,
as well as to relocate an arbitrary line to a general position if the eigenspace is a complex one.
Proof of Theorem 1.9 is given in §6. The main tool used there is Fourier analysis. And the
underlying geometric idea is that the positive entropy in V guarantees that the difference vector
i
between a random pair of nearby points taken with respect to µ is not too short, and by applying
the results obtained in §5 to this vector we can create a sufficiently long line segment placed in a
general position, which is equidistributed on the torus.
In §7 Theorem 1.7 is deduced from 1.9 by taking a test function, we then prove its corollaries.
The appendix §8 gives an application of results in §7 to a number theoretical problem, following a
strategy observed by Cerri [5].
Notation 1.13. Throughout this paper,
• log stands for the logarithm of base 2 and the natural logarithm is denoted by ln.
• For a linear map f, kfk := sup |f(x)| is its norm as an operator.
|x|=1
• If A and B are subsets of some additive group, let A − B (resp. A + B) denote the set
{a−b (resp. a+b) |a ∈ A,b ∈ B}.
• The symbol rank refers to the torsion-free rank of a finitely generated abelian group.
• We denote constants by c ,c ,... and less important ones by κ ,κ ,.... The constants only
1 2 1 2
locally used by a proof are written as ι ,ι ,.... All constants in this paper are going to be
1 2
positive and effective: i.e. an explicit value for it can be deduced from the information already
known if one really want to. When a constant first appears, we may write it as a function to
signify all the variables that it depends on, for example if we write c (N,d), it means that the
3
constant c depends only on N and d, and so forth.
3
4
• We write A . B (or B & A) for the estimate ∃c > 0,A ≤ cB. Moreover, we always include
all factors that the implied constant c depends on within the . symbol as subscripts. For
example, if c depends on and only on d then we will always write A . B; the inequality
d
A . B without any subscript means the implied constant is absolute.
Acknowledgments: This paper is part of my Ph.D. thesis work. I am grateful to my advisor
Prof. Elon Lindenstrauss for introducing me to the subject and guiding me through the research.
I also would like to thank Prof. Jean Bourgain for comments and encouragement.
2 Abelian algebraic actions on the torus
In this section, we are going to impose an alternative condition on G to replace Condition 1.5.
2.1 Preliminaries on number fields
Consider now a degree d number field K, d ≥ 3. K ⊗QR ∼= Rd. Any element of K acts on this
space naturally by multiplication: × .(t⊗x) = st⊗x,∀s,t ∈ K,x ∈ R. Recall that if K has r
s 1
real embeddings σ ,··· ,σ and r pairs of conjugate imaginary embeddings σ ,σ ,j =
1 r1 2 r1+j r1+r2+j
1,··· ,r2, then there is an isomorphism σ : K ⊗QR 7→ Rr1 ×Cr2 ∼= Rd where
(cid:0)
σ(t⊗x) =x· σ (t),··· ,σ (t),Reσ (t),··· ,Reσ (t),
1 r1 r1+1 r1+r2 (2.1)
(cid:1)
Imσ (t),··· ,Imσ (t)
r1+1 r1+r2
for all x ∈ R, t ∈ K. With this identification, the action of K on Rd ∼= Rr1×Cr2 is easy to describe:
×s acts on K⊗QR via the linear mapping (×σ1(s),··· ,×σr1(s),×σr1+1(s),··· ,×σr1+r2(s)), where the
first r multiplications are on the r real subspaces and the last r ones are on the complex ones.
1 1 2
Here we view the j-th copy of C as a two-dimensional real subspace. In addition, let it be spanned
by the (r +j)-th (real part) and the (r +r +j)-th (imaginary part) coordinates: a number z in
1 1 2
this copy of C is identified with (Rez)e +(Imz)e , then the multiplicative action of s on
r1+j (cid:18) r1+r2+j (cid:19)
Reσ (s) −Imσ (s)
this embedded copy of C is given by the matrix r1+j r1+j .
Imσ (s) Reσ (s)
r1+j r1+j
Let O be the ring of integers and U = O∗ be the group of units of K. Dirichlet’s Unit
K K K
Theorem claims that, modulo the torsion part T of U , which is a finite set of roots of unity, the
k k
group morphism
(cid:0) (cid:1)
L : t → log|σ (t)|,log|σ (t)|,··· ,log|σ (t)| (2.2)
1 2 r1+r2
embeds U as a lattice in the (r +r −1)-dimensional subspace
K 1 2
Xr1 Xr2
W = {(w ,w ,··· ,w )| w +2 w = 0} ⊂ Rr1+r2. (2.3)
1 2 r1+r2 j r1+j
j=1 j=1
(cid:26)
1, 1 ≤ i ≤ r ;
Let r = r +r −1 = rank(U ) and d = 1 then d = Pr1+r2d .
1 2 K i 2, r +1 ≤ i ≤ r +r . 1 i
1 1 2
Remark 2.1. Foragivend, thereareonlyfinitelymanypossiblevaluesforr as d−1 = r1+r −1 ≤
2 2 2
r = r +r −1 ≤ r +2r −1 = d−1.
1 2 1 2
5
For an element in K, its logarithmic Mahler measure measures the size of its image under L.
Definition 2.2. For a non-zero element t from a number field K of degree d, the logarithmic
Mahler measure of t is
X
hMah(t) = d log |t| ,
ν + ν
ν
where log x = max(logx,0), ν runs over all finite or infinite places of K and d = [K : Q ],ν|ν˜
+ ν ν ν˜
is the local degree of ν.
hMah(t)
h(t) = is the absolute logarithm height of t, it is determined by the algebraic
d
number t and does not depend on the field K in which t is regarded as an element.
In particular, if t ∈ U then all its non-Archimedean absolute values are equal to 1, so the
K
logarithmic Mahler measure involves only Archimedean embeddings:
hMah(t) = rX1+r2dilog+|σi(t)| = 12 rX1+r2di(cid:12)(cid:12)log|σi(t)|(cid:12)(cid:12) = 21 Xd |logσi(t)| (2.4)
i=1 i=1 i=1
because Pr1+r2d log|σ (t)| = log|Qd σ (t)| = 0.
i=1 i i i=1 i
It is easy to see
hMah(tt0) ≤ hMah(t)+hMah(t0); hMah(tn) = |n|hMah(t),∀n ∈ Z. (2.5)
Definition 2.3. Let U be a finite index subgroup in U . Define the size of (virtually) funda-
K
mental units to be
n o
F := inf F|{hMah(t) ≤ F,t ∈ U} generates a finite-index subgroup of U .
U
Notice as U is a discrete set, F can be achieved, i.e. there are r elements t ,··· ,t that
U 1 r
virtually generate the torsion-free part of U, such that max hMah(t ) = F .
k k U
Remark 2.4. It is known that for all algebraic numbers t of degree d, hMah(t) has an effective
positive lower bound κ (d) depending only on d as long as t is not a root of unity (see Voutier [21]).
1
By definition this is also a lower bound for F .
U
We are interested in lattices embedded by σ into K ⊗Q R, the covolumes and shapes of such
lattices will be important for us.
p
Definition2.5. (1). ThescaleofalatticeΛinad-dimensionalvectorspaceV isS = d covol(Λ).
Λ
(2). The uniformity of a linear isomorphism Ψ : Rd 7→ Rd is
M :=max(kΨkS−1 ,kΨ−1kS )
Ψ Ψ(Zd) Ψ(Zd)
=max(kΨk·|detΨ|−d1,kΨ−1k·|detΨ|d1).
Definition 2.6. A number field K is a CM-number field if it has a proper subfield F such that
rank(U ) = rank(U ).
K F
Actually, it follows from Dirichlet’s Unit Theorem that a number field is CM if and only if K
is a totally complex quadratic extension of a totally real field F (see Parry [18]).
We show a property of non-CM fields that is going to be relevant later.
6
Lemma 2.7. If K is not CM and has a complex embedding σ ,1 ≤ j ≤ r , let F be the proper
r1+j 2
subfield σ−1 (R). If F 6= Q then ∃k,l ∈ {1,··· ,r +r }\{r +j} such that k 6= l but ∀t ∈ F,
r1+j 1 2 1
|σ (t)| = |σ (t)|.
k l
Proof. Let d0 = [K : F] ≥ 2, then each embedding of F extends to exactly d0 different embeddings
of K. In other words, the set of embeddings {σ ,··· ,σ } of K can be divided into d0-tuples, each
1 d
corresponds to one embedding of F; the number of these tuples is [F : Q] = d.
d0
If d0 = 2, each tuple is a pair. As σ and its complex conugate σ has the same
r1+j r1+r2+j
restriction on the maximal real subfield F = σ−1 (R), they form one of the pairs. So any other
r1+j
pairdoesnotcontainσ orσ . Supposeeveryotherpairconsistsoftwoconjugatecomplex
r1+j r1+r2+j
embeddings σ and σ¯ then as they have the same restriction on F, F lies in their common real part.
So all embeddings of F is real and extends to two conjugate complex embeddings of K, which
contradicts the assumption that K is not CM. Hence there is a pair made of two embeddings which
are not complex conjugates to each other.
If d0 ≥ 3, then one of the tuple contains σ and σ . As F 6= Q, there is at least one
r1+j r1+r2+j
other d0-tuple. As d0 ≥ 3, there are two embeddings in that tuple that are not complex conjugate
to each other.
Soinanycasetherearetwodifferentembeddingsσ andσ0 ofK, whicharenotσ orσ
r1+j r1+r2+j
and are not complex conjugate to each other, such that σ| = σ0| . If one or both of them is not
F F
in σ ,··· ,σ then we replace it by its complex conjugate, which does not change the absolute
1 r1+r2
value. This completes the proof.
Condition 2.8. G is an abelian subgroup of SL (Z) such that there are:
d
(1). a non-CM number field K of degree d whose group of units U has rank at least 2;
K
(2). an isomorphim φ from G to a finite-index subgroup of U ;
K
(3). a cocompact lattice Γ in K ⊗Q R which sits in K < K ⊗Q R and is invariant under the
natural action of φ(G),
such that by identifying G with φ(G), the action of G on Td is conjugate to the natural G-action
on X = (K ⊗Q R)/Γ, i.e., there exists a linear isomorphism ψ : Rd 7→ K ⊗Q R ∼= Rd such that
ψ(Zd) = Γ and ψ−1◦× ◦ψ = g,∀g ∈ G.
φ(g)
Remark 2.9. rank(U ) = r +r −1 ≥ 2 implies d = r +2r ≥ 3.
K 1 2 1 2
A consequence to Condition 2.8 is that the logarithmic Mahler measures defined respectly by
Definition 1.6 on G and by Definition 2.2 on φ(G) are identified via φ.
Lemma 2.10. If G satisfies Condition 2.8, then ∀g ∈ G, the logarithmic Mahler measure m(g) =
hMah(φ(g)).
Proof. The eigenvalues of g are exactly those of × , namely σ (φ(g)), ···, σ (φ(g)). By (1.1),
φ(g) 1 d
m(g) = P log σ (φ(g)) = hMah(φ(g)).
i + i
Remark 2.11. If a generating set S of G is fixed, by Dirichlet’s Unit Theorem it is not difficult to
see that the word metric kgkWL of an arbitrary element g with respect to S has some upper bound
kgkWL . hMah(φ(g))+1, from which Remark 1.12 follows.
G,S
It turns out Condition 2.8 is a good substitute for Condition 1.5.
Theorem 2.12. Condition 1.5 implies Condition 2.8.
7
Proof. The theorem decomposes into Propositions 2.13, 2.14 and 2.16, which are proved below.
In fact, it can be shown that the two conditions are equivalent.
2.2 Construction of the number field
InthefollowsingthreesubsectionswearegoingtoproveTheorem2.12. Thefollowingstatement
is a special case of a known theorem (see for example [6, Prop. 2.1], , for a proof of the general
statement see [20]).
Proposition 2.13. Suppose an abelian subgroup G of SL (Z) has an element whose action is
d
irreducible, then there exist
(i). a number field K of degree d;
(ii). an isomorphim φ from G to a subgroup of the group of units U ;
K
(iii). a rank d lattice Γ in K < K ⊗QR which is invariant under the natural action of φ(G),
such that there exists a linear isomorphism ψ : Rd 7→ K ⊗Q R ∼= Rd such that ψ(Zd) = Γ and
ψ−1◦× ◦ψ = g,∀g ∈ G;
φ(g)
Proof of Proposition. Assume g ∈ G acts irreducibly, then the characteristic polynomial of g is
irreducible over Q; otherwise the rational canonical form of g over Q has more than one block,
giving a non-trivial g-invariant rational subspace of Rd, which projects to a non-trivial g-invariant
subtorus in Td. Therefore g has d distinct eigenvalues ζ1,··· ,ζd which are algebraic conjugates to
g g
each other, where ζg1,··· ,ζgr1 are real and the rest are r2 imaginary pairs, r1 +2r2 = d. ζgr1+j =
ζr1+r2+j,∀j = 1,··· ,r .
g 2
Construct number field K = Q(ζi), so degK = d. Denote K = K , then K has d embeddings:
i g i 1
σ (ζ1) = ζi,σ (K) = K ,∀i = 1,··· ,d. The first r embeddings are real; σ and σ are
i g g i i 1 r1+j r1+r2+j
complex conjugates for 1 ≤ j ≤ r .
2
An eigenvector v of ζi lies in Ker(g−ζ1id). As all entries of g−ζ1id belongs to the field K,
1 g g g
we can fix an eigenvector v ∈ Kd.
1
Let v = σ (v ) ∈ K d, then as g ∈ SL (Z) is fixed by σ , gv = σ (g)σ (v ) = σ (ζ1v ) =
i i 1 i d i i i i 1 i g 1
σ (ζ1)σ (v ) = ζiv . So v is an eigenvector corresponding to the eigenvalue ζi for all 1 ≤ i ≤ d and
i g i 1 g i i g
it follows that Cn = ⊕d Cv . Notice v ∈ σ (Kd) = Kd ⊂ Cd, in particular it is a real vector for
i=1 i i i i
1 ≤ i ≤ r .
1
As g is irreducible all eigenspaces are one dimensional over C, thus by commutativity the basis
{v }d diagonalizes not only g but any element h ∈ G as well: hv = ζiv ,∀h ∈ G,∀i.
i i=1 i h i
We claim ζi ∈ K ,∀h ∈ G. This is because an arbitrary element t in the field Q(ζi,ζi) can be
h i g h
written as p(ζi,ζi) where p is a rational polynomial. As gh = hg, p(g,h)v = p(ζi,ζi)v = tv , so
g h i g h i i
t is an eigenvalue of the rational d×d matrix p(g,h), thus an algebraic number of degree at most
d. By choosing t to be a generating element of Q(ζi,ζi) = K (ζi), degK (ζi) ≤ d = degK . Thus
g h i h i h i
ζi ∈ K .
h i
Furthermore, σ (ζ1) = ζi for any h ∈ G. Actually, as ζ1 ∈ K, ζ1 = f(ζ1) for some polynomial
i h h h h g
f ∈ Q[x]. f(g)v = f(ζ1) = ζ1v = hv . If f(g) 6= h then 0 < rank(f(g)−h) < d, Ker(f(g)−h)
1 g h 1 1
is a non-trivial rational subspace of Rd. Since g commutes with f(g)−h, this rational subspace
is g-invariant, which is prohibited to happen as g acts irreducibly. So h = f(g), in particular
σ (ζ1) = σ (f(ζ1)) = f(σ (ζ1)) = f(ζi) = ζi.
i h i g i g g h
8