On the rate of accumulation of $\alpha\zeta^{n}$ mod 1 to 0

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by  Johannes Schleischitz

ON THE RATE OF ACCUMULATION OF (αζn)n≥1 MOD 1 TO 0
JOHANNES SCHLEISCHITZ
Abstract. In this paper we study the distribution of the sequence (αζn) mod 1,
n 1
≥
where α,ζ are fixedpositive realnumbers,with specialfocus on the accumulationpoint
4 0. For this purpose we introduce approximation constants σ(α,ζ),σ(α,ζ) and study
1 their properties in dependence of α,ζ, distinguishing in particular the cases of Pisot
0
numbers, algebraic non Pisot numbers and transcendental values of α as well as ζ.
2

n
a
J Supported by FWF grant P24828

9 Institute of Mathematics, Department of Integrative Biology, BOKU Wien, 1180,Vienna, Austria
2

] Math subject classification: 11J71,11J81, 11J82,11K55
T
key words: Pisot numbers, distribution mod 1, Diophantine approximation,transcendence theory
N
.
h
t
a
m
1. Introduction
[

1 This paper deals with the distribution of αζn mod 1 for arbitrary but fixed positive real numbers α,ζ
v
as n runs through the positive integers. We are in particular interested in pairs α,ζ for which rather
8
8 fast accumulation to 0 occurs, either for a sequence of arbitrarily large values of n or for all sufficiently
5 large values of n. We will treat these two cases separately, measuring the rate of accumulation with
7 approximation constants σ(α,ζ),σ(α,ζ) we will introduce in section 1.1. Related problems were first
.
1 studied by Pisot in [21] using methods of Fourier Analysis. An interesting result of Pisot states that if
0 for some ζ > 1 the sequence αζn mod 1 tends to zero as n tends to infinity, so roughly speaking the
4 numbers αζn somehow ”converge to integers”, then ζ must be algebraic and of a special shape, called
1
Pisot numbers to his honors. We will give a definition and known properties of Pisot numbers in the
:
v section 1.1.
i
X
In the present paper we don’t make use of Fourier analysis, many given proofs rely on basic properties
r of symmetric polynomials or classical Diophantine approximation properties, in the latter case mostly
a
concerning the approximation of n ζ mod 1 for fixed ζ R as n runs through the integers, and higher
· ∈
dimensionalgeneralizations. Forthispurposeweatfirstintroducesomenotation,someofwhichisclassic
notation and some invented for our special purpose.
1.1. Basic facts and notations. At first a basic definition, whose parameter x will later mostly be of
the form αζn.
Definition 1.1. For a real number x R denote with x Z the largest integer smaller or equal x,
∈ ⌊ ⌋ ∈
x Zthesmallestintegergreaterorequalxand x [0,1)thefractionalpartofx,i.e. x =x x .
⌈ ⌉∈ { }∈ { } −⌊ ⌋
Furthermoredenotewith x Ztheclosestintegertoxandwith x := x x [0,1/2]thedistance
h i∈ k k | −h i|∈
to the closest integer to x, with the special convention if x =1/2 then x := x . So clearly we have
{ } h i ⌊ ⌋
x x , x and x = x x . If for a sequence (x ) we have lim x = 0, we will say
n n 1 n n
h i ∈ {⌊ ⌋ ⌈ ⌉} k k | −h i| ≥ →∞k k
(x ) converges to integers.
n n 1
≥
1
2 JOHANNESSCHLEISCHITZ
By Bolzano-Weierstrass Theorem [12], an alternative characterization of convergence to integers is that
the sequence x can only have the accumulation point 0 .
n
{ } { }
We will in general restrict to the case
(1) ζ >1, α>0,
as for ζ ( 1,1) we clearly have lim αζn = 0 for all α R and ζ ζ, α α does not affect
n
the prop∈erti−es of αζn mod 1, and the s→pe∞cial cases ζ 1,1∈ or α=07→are−of no7→inte−rest either.
∈{− }
We are particularly interestedin α,ζ, for which at least for a subset (n ,n ,...) of positive integers, the
1 2
values αζni converge to integers rather quickly.
In order to measure this convergence and more general the distribution of αζn mod 1 in dependence of
n, we now introduce
log αζn
(2) σ (α,ζ):= k k
n − log(αζn)
such as the derived approximation constants
(3) σ(α,ζ):=liminfσ (α,ζ), σ(α,ζ):=limsupσ (α,ζ).
n n
n→∞ n→∞
Large values of σ(α,ζ) mean that for some sequence (n ,n ,...) of positive integers which tends mono-
1 2
tonically to infinity, the values αζni converge to integers very fast. In particular, σ(α,ζ)>0 gives an
k k
exponential convergence to integers of the sequence.
Similarly, large values of σ(α,ζ) give fast convergence of the sequence (αζn) to integers, and in
n 1
≥
particular σ(α,ζ)>0 gives exponential convergence.
In case of α=0 and ζ ( 1,1), it is easy to see that
6 ∈ −
(4) σ(α,ζ)=σ(α,ζ)= 1, ζ ( 1,1).
− ∈ −
So in the sequelassume ζ >1,α>0. In this casewe havelim αζn = as wellas 0 αζn 1/2
n
→∞ ∞ ≤k k≤
for all n, so clearly
(5) σ(α,ζ) σ(α,ζ) 0, ζ >1.
≥ ≥
Note that the expressions σ(α,ζ),σ(α,ζ) can be written in the easier form
log αζn log αζn
(6) σ(α,ζ)=liminf k k, σ(α,ζ):=limsup k k.
n→∞ − n·logζ n→∞ − n·logζ
This can easily be deduced by the definition of the quantities, as for sequences (x ) ,(y ) with
n n 1 n n 1
lim y = as well as lim xn =:Z and fixed δ R we have ≥ ≥
n→∞ n ∞ n→∞ yn ∈
x x y
n n n
(7) lim = lim =Z 1=Z.
n→∞yn+δ n→∞ yn · yn+δ ·
Applying (7) to x := log αζn ,y := logζn = nlogζ and δ = logα, which satisfy the conditions by
n n
− k k
(1), and recalling (3) yields (6).
Also note that in the case σ(α,ζ) = 0 respectively σ(α,ζ) = 0, one cannot decide if the sequence αζn
respectively some subsequence αζni tends to integers.
An easy property of the quantities σ (α,ζ) is σ (α,ζk) = σ (α,ζ) for all α,ζ and k = 1,2,3,... and
n n nk
thus taking limits
(8) σ(α,ζk) σ(α,ζ), k =1,2,3,...
≥
(9) σ(α,ζk) σ(α,ζ), k =1,2,3,...
≤
holds. Anothereasypropertyofthequantitiesσ(α,ζ),σ(α,ζ)isgiveninthefollowingPropositionwhich
we will use in section 2.2.2.
ON THE RATE OF ACCUMULATION OF (αζn)n≥1 MOD 1 TO 0 3
Proposition 1.2. Let α,ζ >1 be real numbers and M,N >0 integers. Then
logζ σ(α,ζ) logN
σ(Mα,Nζ) max · − ,0
≥ logζ+logN
(cid:18) (cid:19)
logζ σ(α,ζ) logN
σ(Mα,Nζ) max · − ,0 .
≥ logζ+logN
(cid:18) (cid:19)
Proof. WithoutlossofgeneralitywemayassumeM,N,α,ζ alltobe positive. Thebound0isthe trivial
bound from (5), so we may restrict to the case logN <σ(α,ζ) resp. logN <σ(α,ζ), or equivalently
logζ logζ
(10) N <ζ−σ(α,ζ), resp. N <ζ−σ(α,ζ).
For any positive integers M,N we have
(Mα)(Nζ)n = (Mα)(Nζ)n (Mα)(Nζ)n
k k |h i− |
= MNnαζn MNnαζn
|h i− |
= MNn( αζn αζn ) MNnαζn
|h h i±k k i− |
= MNn αζn MNn αζn MNnαζn
| h i±h k ki− |
= MNn( αζn αbn) MNn αζn
| h i− ±h k ki|
(11) = MNn αζn MNn αζn .
|± k k±h k ki|
By our restrictions (10), for any sufficiently large n n resp. for arbitrarily large values of n n we
0 0
≥ ≥
have MNn αζn < 1, which is equivalent to MNn αζn =0. In view of (11) this yields
k k 2 h k ki
(Mα)(Nζ)n MNn αζn
k k≤ k k
for the respective values n. Taking logarithms according to (2) yields for any ǫ > 0 and the respective
values of n (restricting to n n =n (ǫ) for some n (ǫ)>n if needed)
1 1 1 0
≥
(σ (α,ζ) ǫ)logζ logN
n
σ (Mα,Nζ) − − .
n
≥ logζ+logN
The assertion follows with ǫ 0 by the definition of the quantities σ(α,ζ),σ(α,ζ). (cid:3)
→
Remark 1.3. Note that in case of α=0 and ζ (0,1) we have
6 ∈
Nζ > 1 = σ(Mα,Nζ) 0
⇒ ≥
Nζ < 1 = σ(Mα,Nζ)=σ(Mα,Nζ)= 1.
⇒ −
This is easily deduced by (4) and (5).
We will now give the definition of a class of algebraic numbers with a highly non-generic and thus
interesting behavior concerning the sequence (αζn) mod 1.
n 1
≥
Definition 1.4. (Pisot numbers, Pisot polynomials, Pisotunits) A realalgebraic integer ζ >1 is called
Pisot number,ifallitsconjugatesliestrictlyinsidetheunitcircleofthecomplexplane. IfaPisotnumber
is a unit in the ring of algebraic integers,we will call it a Pisot unit. We will refer to the monic minimal
polynomial P Z[X] of a Pisot number ζ as the Pisot polynomial of ζ. In general call a polynomial a
∈
Pisotpolynomialif itis the Pisotpolynomialof a Pisotnumber ζ, andthe unique rootgreaterthan 1 of
a Pisot polynomial P the Pisot number associated to P.
We will sum up known results for Pisot numbers we will refer to in the sequel, transferred into our
notation, in the following Theorem (which we will call Pisot Theorem although the results may not be
entirely due to him).
4 JOHANNESSCHLEISCHITZ
Theorem 1.5 (Pisot). Pisot numbers have the property σ(1,ζ) > 0, i.e. ζn converges to integers at
exponential rate. This property characterizes Pisot numbers among all real algebraic numbers. Even the
following stronger assertion holds: if αζn tends to integers for a real algebraic number ζ > 1, then ζ is
a Pisot number and α Q(ζ), where α=1 is always a possible choice.
∈
Moreover, if for any real ζ > 1 the sequence ( αζn ) is square-summable, then ζ is a Pisot number
n 1
k k ≥
(and clearly again α Q(ζ), and α=1 is always a possible choice.)
∈
The first assertion is easily seen by looking at the sum of the powers k ζn of the conjugates ζ =
j=1 1
ζ,ζ ,...,ζ of ζ, where k denotes the degree [Q(ζ) : Q] of ζ. Every such sum is an integer as it is
2 k
P
a symmetric polynomial in ζ ,ζ ,...,ζ . We will recall a detailed proof in Theorem 2.5. See [21] or
1 2 k
chapter 5 in [3] for the proofs of the remaining and slightly refined results. At this point it should be
mentioned that there are only countably many ζ such that αζn converges to integers for some auxiliary
α, an immediate consequence of Theorem 5.6.1 in [3], but the question if any such ζ is transcendentalis
open. Theorem 1.5 immediately yields
Theorem 1.6. Let ζ > 1 be a real number but not a Pisot number. Then for any α R we have
∈
σ(α,ζ)=0. If ζ is a Pisot number and α / Q(ζ), we have σ(α,ζ)=0 as well.
∈
Proof. If otherwise αζn ζ nǫ for some ǫ>0 and all n n sufficiently large, then
− 0
k k≤ ≥
∞ n0−1 ∞ 1
αζn 2 αζn 2+ ζ 2nǫ n + < .
k k ≤ k k − ≤ 0 1 ζ 2ǫ ∞
nX=1 nX=1 nX=n0 − −
This contradicts the fact that only Pisot numbers have this property by Theorem 1.5. The proof of the
second assertion is similar due to facts from Theorem 1.5. (cid:3)
We will discuss properties of Pisot numbers concerning the approximation constants σ(1,ζ),σ(1,ζ) in
more detail in section 2.1. Now we will only give one more well known basic fact
Proposition 1.7. Any Pisot polynomial P is irreducible.
Proof. Clearly, the constant coefficient of P is a nonzero integer. Consequently, if P = Q R, with
·
Q,Rnon-constantpolynomials,theconstantcoefficientsofQ,Rhavethispropertytoo,sotheirabsolute
values are at least 1. Hence both Q,R must have at least one root of absolute value larger than 1, since
by Vieta Theorem 2.4 the product of the roots of Q resp. R are just the constant coefficient of Q resp.
R. A contradictionto the fact that there is only one rootofP with absolute value greateror equalthan
1. (cid:3)
As indicated in the introduction section 1, our approach will at some places deal with a classic simulta-
neous Diophantine approximationproblem.
Definition 1.8. For a positive inter d and ζ :=(ζ ,ζ ,...,ζ ) Rd define by λ (ζ) respectively λ (ζ)
1 2 d d d
∈
the supremum of all µ R such that
∈
b
x X
| | ≤
(12) max xζ y X µ
i i −
1 k d| − | ≤
≤ ≤
has a solution (x,y ,...,y ) Zd+1 for some arbitrarily large values of X respectively all sufficiently
1 d
∈
large values of X.
By Minkowski’s lattice point Theorem, for all ζ Rd we have the well known result
∈
1
(13) λ (ζ) λ (ζ) ,
d d
≥ ≥ d
seethefirstpagesof[22]forinstance. Foralmostallζ Rd,thereisactuallyequalityinbothinequalities
b
∈
(13), see [14]. For our purposes it suffices to restrict to the case d=1.
ON THE RATE OF ACCUMULATION OF (αζn)n≥1 MOD 1 TO 0 5
Theorem 1.9 (Khinchin). The set of ζ R with λ (ζ)>1 has Lebesgue measure 0.
1
∈
We will later need the following result by Davenport, Schmidt and Laurent, see [7],[15].
Theorem 1.10 (Davenport,Laurent,Schmidt). Let ζ =(ζ,ζ2,...,ζd) for ζ R not algebraic of degree
∈
d . Then
≤ 2
(cid:6) (cid:7)
1
λ (ζ) .
d ≤ d
2
b (cid:6) (cid:7)
We will also refer to the well-known Roth Theorem for algebraic numbers which we will state in our
notation.
Theorem 1.11 (Roth). For an irrational, algebraic real number ζ we have λ (ζ)=1.
1
We will later refer to another type of classical approximation constants too that deal with linear forms
and that are somehow dual to λ and λ .
d d
Definition 1.12. For a real vector ζ = (ζ ,ζ ,...ζ ) let the quantities w (ζ) resp. w (ζ) be given by
b 1 2 d d d
the supreme of all ν R such that
∈
b
(14) x+y ζ +y ζ + +y ζ H ν
1 1 2 2 d d −
| ··· |≤
has a solution (x,y ,...,y ) Zd+1 with max(x, y ,..., y ) H for arbitrarily large resp. all
1 d 1 d
∈ | | | | | | ≤
sufficiently large values H.
A connection between the values of λ and w is given by
d d
Theorem 1.13 (Khinchin). For any ζ Rd we have
∈
w (ζ)
w (ζ) dλ (ζ)+d 1, λ (ζ) d .
d ≥ d − d ≥ (d 1)w (ζ)+d
d
−
Finally a definition about polynomials we will use frequently in section 2.
Definition 1.14. For a polynomial P(X) = a Xk +a Xk 1 + +a with integer coefficients let
k k 1 − 0
− ···
H(P) := max a . For an algebraic number z put H(z) := H(P) with the minimal polynomial
0 j k k
≤ ≤ | |
with relatively prime coefficients P Z[X] of z.
∈
Note that the quantities w (ζ),w (ζ) can be equivalently defined as
d d
logL(ζ) logL(ζ)
(15) w (ζ)=limsup bmax , w (ζ)=liminf max ,
d d
H→∞ kLk∞≤H− logH H→∞ kLk∞≤H− logH
where L(X ,X ,...,X ) = a +a X +a X + +ba X for a Z and L := max a . So
1 2 d 0 1 1 2 2 d d j 0 j d j
··· ∈ k k∞ ≤ ≤ | |
for any H > 0 the maxima are taken among all integral linear forms L with coefficients bounded by
H. Sprindzuk [25] works with a similar notation for example. The expressions in (15) are in notable
conformity to the possible definition of the quantities σ(α,ζ),σ(α,ζ) in (6), which againunderlines that
it is pretty natural to consider these quantities.
6 JOHANNESSCHLEISCHITZ
2. Results for σ(α,ζ),σ(α,ζ) with algebraic α,ζ
2.1. Pisot numbers. Pisotdiscoveredin that the sequence of positive integer powersof Pisot numbers
converge exponentially to integers, i.e. σ(1,ζ) > 0. We want to prove some more detailed results in
terms of the quantities σ(1,ζ),σ(1,ζ). For preparation we need the following well-known result.
Definition 2.1. A polynomial P Z[X ,...,X ] is called symmetric, if for all bijections (=permuta-
1 k
∈
tions) σ : 1,2,...,k 1,2,...,k the polynomial remains unaffected.
{ }7→{ }
Definition 2.2. The elementary symmetric polynomials in k variables X ,X ,...,X are given by
1 2 k
k
µ := X , µ := X X , ..., µ := X .
k,1 j k,2 i j k,k j
j=1 1 i<j k 1 j k
X ≤X≤ ≤Y≤
Theorem 2.3. Every symmetric polynomial is a polynomial with integer coefficients in the elementary
symmetric polynomials.
See [16] for a proof.
Theorem 2.4(Vieta). LetP =a Xk+a Xk 1+ +a beapolynomial withintegercoefficients and
k k 1 − 0
roots ζ1,...,ζk counted with multiplicity. T−hen aj = ·(−··1a)kk−jµk,k+1−j with µ.,. from the above Definition.
Now we are ready to present a first Theorem concerning the quantities σ(1,ζ),σ(1,ζ).
Theorem 2.5. Let ζ be a Pisot number of degree [Q(ζ):Q]=k. Then we have
1
(16) 0 < σ(1,ζ)
≤ k 1
−
(17) 0 < σ(1,ζ) σ(1,ζ) k 1
≤ ≤ −
Proof. Let ζ =ζ,ζ ,...,ζ be the conjugates of ζ. Note first, that for all positive integers n
1 2 k
k
(18) ζn Z, n 1.
j ∈ ≥
j=1
X
Thatisbecauseitisasymmetricpolynomialinthevariablesζ withintegercoefficients. ByTheorem2.3
j
it is an integer linear combination of the elementary symmetric polynomials, which are itself integers as
theyarethecoefficientsoftheminimalpolynomialP(X):= k (X ζ )ofζ (observeζ isanalgebraic
j=1 − j
integer). Thus we have (18).
Q
We now first proof (16). On the other hand, by definition all other roots ζ ,ζ ,...,ζ of P(X) apart
2 3 k
fromζ haveabsolutevalue smallerthan1,solet0<f <1be the maximumabsolutevalueamongthese
and put z := logf <0. Then for all positive integers n we have
logζ
k
(19) ζn (k 1) fn =(k 1) ζnz.
j ≤ − · − ·
j=2
X
On the other hand, by (18) and as the right hand side of (19) converges to 0 as n we have
→∞
k
(20) ζn = ζn, n n .
k k j ≥ 0
j=2
X
Combining (20) with (19) we have
log ζn k ζn log((k 1)ζnz)
j=2 j
σ(1,ζ):=liminf k k =liminf liminf − = z >0,
n −n logζ n − n logζ ≥− n n logζ −
→∞ · →∞ P· →∞ ·
ON THE RATE OF ACCUMULATION OF (αζn)n≥1 MOD 1 TO 0 7
the left hand side of (16).
For the upper bound in (16) first observe, that since the product of the roots is the constant coefficient
of P(X) which is a nonzero integer,
k k
1 ζ =ζ ζ ζ fk 1
j j −
≤ | | ≤ ·
j=1 j=2
Y Y
1
yields f ζ−k−1 or equivalently
≤
1
(21) z .
≤ k 1
−
Now let ψ = 0,ψ ,...,ψ be the angles of ζ in the complex plane in the interval [0,2π) and put
1 2 k k
φ := ψj [0,1)for 1 j k. In generaldenote by ψ(t) the anglein [0,2π) ofa complex number t and
j 2π ∈ ≤ ≤
put φ(t):= φ(t). By the identity ψ(tn)=n ψ(t) we have
2π ·
(22) ψ(ζn) = n ψ , 1 j k
j · j ≤ ≤
(23) φ(ζn) = n φ , 1 j k.
j · j ≤ ≤
By (13) with d=n there exist arbitrarily large values of n such that simultaneously
1 1
φ(ζn) = nψ < 2π, 1 j k
k j k k jk≤ n 8 · ≤ ≤
for certain arbitrarily large values of n. (For the rest of of the proof of (16) we only consider such a
sequenceofvaluesforn,andassumenissufficientlylarge.) Henceψ(ζn)=2π φ(ζn),wheretheequality
j · j
is viewed mod 2π, hence φ(ζn) 2π < π.Thus
| j − | 4
1
Re ζn cos(ψ) ζn =cos(ψ) ζ n ζ n, 1 j k.
j ≥ j j j | j| ≥ √2 ·| j| ≤ ≤
Since k ζn is an inte(cid:0)ger(cid:1)and ζn is r(cid:12)(cid:12)eal(cid:12)(cid:12), k ζn is real. Hence and recalling (20) gives
j=1 j j=2 j
P k P k k
1 1
ζn = ζn =Re ζn ζ n fn.
k k j  j ≥ √2 · | j| ≥ √2 ·
j=2 j=2 j=2
X X X
 
Recalling (21) and taking logarithms yields the right hand side of (16).
For the remaining non trivial upper bound of (17) put M := ζn Z and consider
n
h i∈
k
Φ(M ):= M ζn .
n n− j
j=1
Y(cid:0) (cid:1)
The values φ(M ) are integers by Theorem 2.3, as it is again a symmetric polynomial in the variables
n
ζ with integer coefficients. Moreover Φ(M ) = 0 for all positive integers n. Indeed, if otherwise ζ
j n
6
would be of the shape ζ = m√L for some L∈Z, but then its conjugates would be ηmj ζ with ηm :=e2mπi,
contradicting the fact that the conjugates apartfrom ζ itself lie inside the unit circle. Clearly, powersof
the other conjugates are smaller than 1 in absolute value so they cannot equal M either.
n
Thus we have
k k
1 M ζn = M ζn M ζn ,
≤ n− j | n− | n− j
j=1 j=2
Y(cid:12) (cid:12) Y(cid:12) (cid:12)
which leads to (cid:12) (cid:12) (cid:12) (cid:12)
1
(24) ζn = M ζn .
k k | n− |≥ k M ζn
j=2 n− j
However, Mn =ζn+o(1) and |ζjn|=o(1) for 2≤j ≤Qk as n(cid:12)(cid:12)→∞. U(cid:12)(cid:12)sing this in (24) gives
(25) ζn (ζn+o(1)) (k 1), n .
− −
k k≥ →∞
8 JOHANNESSCHLEISCHITZ
Now the fact σ(1,ζ) cannot exceed k 1 follows easily. If otherwise σ(1,ζ) > k 1, then for some
− −
arbitrarily large values of n and some ǫ>0 we would have
ζn ζ−n(1+ǫ)(k−1) = ζ1+ǫ −n(k−1) =ζ−n(k−1)
k k≤
with ζ :=ζ1+ǫ >ζ, so lim ζn ζn = , clearly(cid:0) cont(cid:1)radicting (25) for n sufficiently large. (cid:3)
n→∞ − ∞ b
Remark 2.6. A generalization for the upper bound in (17) for σ(α,ζ) for arbitrary algebraic real
b b
numbers α is given in Corollary 2.33. An interesting question is if in fact σ(α,ζ) = 0 for all Pisot
numbersζ andalgebraicnumbersα / Q(ζ)andmoregeneralforanyalgebraicζ andalgebraicα / Q(ζ).
∈ ∈
For transcendental α this is wrong, see Theorem 3.5.
For the quantity σ(α,ζ) with ζ a Pisot number and real α / Q(ζ) we have σ(α,ζ)=0 by Theorem 1.6.
∈
Analyzing the proof of Theorem we obtain a Corollary that we will refer to in the sequel.
Corollary 2.7. For a Pisot number ζ of degree [Q(ζ):Q]=k with conjugates ζ ,ζ ,...,ζ we have
2 3 k
logf
σ(1,ζ)
≤−logζ
where f :=max ζ <1.
2 j k j
≤ ≤ | |
Proof. The proof of the right hand side of (16) effectively contained the proof of the claim. (cid:3)
The upper bound for σ(1,ζ) seems to be not very strong. It seems unlikely that the case σ(1,ζ)> 1 ,
k 1
or more general −
(26) σ(1,ζ)>σ(1,ζ),
does occur for any Pisot number ζ. A stronger result should be true.
Conjecture 2.8. For any Pisot number ζ and any real α we have σ(α,ζ)=σ(α,ζ).
We want to give a result that somehow quantifies this, which connects the constant σ(1,ζ) with the
Diophantine problem (12) for some ζ arising from the conjugates of ζ. First we define
Definition 2.9. Denote with Liov the set of Liouville numbers, which is irrationalrealnumbers ζ such
that λ (ζ)= .
1
∞
It is known that Liov consists only of transcendental numbers, as irrational algebraic numbers ζ have
λ (ζ)=1 by Roth‘s Theorem, that can be found in [4], and (13).
1
Liov has Hausdorff dimension 0, which is an easy consequence of the following Theorem by Bernik [2]:
Theorem 2.10 (Bernik). For apolynomial P let H(P)be the maximum absolute value of its coefficients
and let
A(w):= ζ R: P(ζ) <H(P)−w for infinitely many P Z[X], deg(P) n
∈ | | ∈ ≤
Then the Hausdo(cid:8)rff dimension of A(w) equals n+1. (cid:9)
w+1
The special case n=1 proves the claim concerning the Hausdorff dimension. So the set Liov is small in
some sense, which is of interest with respect to our following Theorem 2.13. For its proof we need
Theorem2.11(Smyth). Letζ beaPisotnumberandζ =ζ,ζ ,...,ζ beitsconjugates. Then ζ = ζ
1 2 k i j
| | | |
for i=j implies ζ =ζ .
i j
6
See [24] for a proof. We will also need a closely related deeper result.
Theorem 2.12 (Mignotte). For a Pisot number, there is no nontrivial multiplicative relation between
its conjugates.
ON THE RATE OF ACCUMULATION OF (αζn)n≥1 MOD 1 TO 0 9
See [18] for a more precise definition and a proof.
Theorem 2.13. Let ζ = ζ be a Pisot number of degree [Q(ζ) : Q] = k. Further let the conjugates
1
ζ = ζ,ζ ,...,ζ be labeled by decreasing absolute values, such that in particular ζ = max ζ .
1 2 k 2 2 j k j
Put ζ2 =R2eiψ2 with 0 ψ2 <2π. | | ≤ ≤ | |
≤
Suppose σ(1,ζ)>σ(1,ζ), which by Theorem 2.5 in particular holds if σ(1,ζ)> 1 . Then the following
k 1
holds −
ψ
2
Liov.
2π ∈
Proof. First note that by Smyth’s Theorem 2.11 the conjugate ζ is determined by the absolute value
2
propertyupto complexconjugation. Define r by ζ =r ζ =r R for3 j k. Putζ =ζ incase
j j j 2 j 2 3 2
| | | | ≤ ≤
of non-real ζ . Then, clearly 0 < r r ... r < 1 in case of real ζ as well as 0 < r r
2 k k 1 3 2 k k 1
≤ − ≤ ≤ − ≤
... r <r =1 in case of non-real ζ by Smyth’s Theorem 2.11.
4 3 2
≤
In view of this, if ζ is real, i.e. ψ =0, for n sufficiently large
2 2
k k
1
(27) ζn = ζn ζ n 1 rn ζ n, n n .
k k (cid:12) j(cid:12)≥| 2|  − j≥ 2 ·| 2| ≥ 0
(cid:12)j=2 (cid:12) j=3
(cid:12)X (cid:12) X
(cid:12) (cid:12)  
Hence and by Corollary 2.7 t(cid:12)his cas(cid:12)e clearly yields σ(1,ζ)=σ(1,ζ) log|ζ2| 1 .
(cid:12) (cid:12) ≤− logζ ≤ k 1
−
If otherwise ψ =0, we similarly have
2
6
k k
(28) ζn = ζn ζn 2cos(nψ ) rn ζ n(2cos(nψ ) (k 3)rn), n n .
k k (cid:12) j(cid:12)≥(cid:12) 2  | 2 |− j(cid:12)≥| 2| 2 − − 4 ≥ 0
(cid:12)j=2 (cid:12) (cid:12) j=4 (cid:12)
(cid:12)X (cid:12) (cid:12) X (cid:12)
Thus by Corolla(cid:12)(cid:12)ry 2.7 a(cid:12)(cid:12)nd(cid:12)(cid:12)noting (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
1+r
4
(29) 0<r < <1,
4
2
if σ(1,ζ)>σ(1,ζ)= log|ζ2|, we must have a sequence of values n such that
− logζ
1+r n
4
(30) cos(nψ )=o , n .
2
2 →∞
(cid:18)(cid:18) (cid:19) (cid:19)
To avoid subindices we only consider this sequence of values n in the sequel. Using twice the addition
Theorem cos(2a) = cos2(a) sin2(a) = 2cos2(a) 1 for cosine we infer cos(4nψ ) = 8cos4(nψ )
2 2
− − −
4cos2(nψ )+1=1+o 1+r4 2n as n . Hence
2 2 →∞
(cid:16)(cid:0) (cid:1) (cid:17) 1+r n
(31) sin(4nψ )= 1 cos2(4nψ )=o 4 , n .
2 2
− 2 →∞
(cid:18)(cid:18) (cid:19) (cid:19)
p
But by (29) the right hand side of (31) tends to 0. Clearly, this implies
(32) 4nψ =2πm +o(1)
2 n
as n for an integer sequence (m ) . Using 1 x sin(x+2mπ) for an integer m and x
→ ∞ n n≥1 2| | ≤ | | ∈
[ 0.1,0.1], (31) yields
− 1+r n
4
4nψ =2m π+o
2 n
2
(cid:18)(cid:18) (cid:19) (cid:19)
too for the sequence in (30) and the corresponding integer sequence (m ) from (32). But again by
n n 1
(29), this in particular implies 4nψ = 2m π +o(n ν) or equivalently (4≥n) ψ2 m = o(n ν) for
2 n − · 2π − n −
any ν > 0 and n sufficiently large. By definition this implies ψ2 is either nonzero rational (the case
2π
of a real numbers ζ was treated in (27)) or a Liouville number. To see that it cannot be rational we
2
use Mingnotte’s Theorem 2.12. If ψ2 is nonzero rational, then ζ would be a real multiple of a root of
2π 2
10 JOHANNESSCHLEISCHITZ
unity. In this case for some positive integer L we would have ζ2L = ζ 2L = ζLζ L, so ζL = ζ L, but
2 | 2| 2 2 2 2
this clearly is a nontrivial multiplicative relation between the roots ζ ,ζ =ζ contradicting Mignotte’s
2 3 2
Theorem. (cid:3)
Theorem 2.13 in fact even shows the stronger condition
φ
n 2 e−νn
2π ≤
(cid:13) (cid:13)
(cid:13) (cid:13)
hasinfinitely many integralsolutions nfor so(cid:13)meν(cid:13)>0 incaseofσ(1,ζ)>σ(1,ζ). However,asalgebraic
numbers are only countable and angles ψj in(cid:13)Theo(cid:13)rem 2.13 are typically expected to be transcendental,
2π
the pathological case cannot be excluded easily. However, we give another Corollary affirming it should
not happen.
Corollary 2.14. If for any real θ in the splitting field of a Pisot polynomial the value arctan(θ) is not
π
a Liouville number, then σ(1,ζ) = σ(1,ζ) for any Pisot number ζ. In particular, if for any algebraic
numberθ thevalue arctan(θ) iseitherrationalorirrationalandnoLiouvillenumber,thenσ(1,ζ)=σ(1,ζ)
π
for any Pisot number ζ.
Proof. For any algebraic number γ =reiψ, since the conjugate is algebraic in the same splitting field so
areRe(γ)= γ+2γ,Im(γ)= γ2−iγ andhence their quotient i(γγ−+γγ) =:θ =tan(ψ)∈R too. Sowemayapply
Theorem 2.13 using its notation with γ =ζ (or equivalently θ =tan(ψ ) or arctan(θ)=ψ ). (cid:3)
2 2 2
Remark 2.15. The condition in Corollary 2.14 is equivalent to the condition that w (π,tan(ψ ))=
2 2
∞
forthetwo-dimensionalsimultaneousapproximationconstantw in(14). ItisworthnotingbyKhinchin’s
2
transference principle Theorem 1.13 this implies λ (π,tan(ψ )) 1.
2 2
≥
The next Proposition shows that indeed for k =2, we cannot have (26).
Theorem 2.16. Let ζ be a Pisot number of degree [Q(ζ) : Q] = k = 2. Then σ (1,ζ) is eventually
n
constant. In particular σ(1,ζ)=σ(1,ζ).
Furthermore the set
(33) t R: ζ Pisot number of degree k=2 such that σ(1,ζ)=σ(1,ζ)=t
{ ∈ ∃ }
is dense in (0,1]. Moreover
(34) σ(1,ζ)=σ(1,ζ)=1
if and only if ζ is an algebraic unit.
Proof. In the quadradic casethe only conjugate ζ ofζ is real,so for n n largeenoughthat ζ n < 1
1 ≥ 0 | 1| 2
we have ζn = ζ n, thus by definition
1
k k | |
log ζ
1
(35) σ (1,ζ)= | | (0,1], n n
n 0
− logζ ∈ ≥
For the second statement first consider only ζ of the shape ζ = N +√d and observe that for positive
integersN,dthe conditions (N 1)2+1 d (N+1)2 1,d=N2, areeasilyseento be necessaryand
− ≤ ≤ − 6
sufficient for such ζ to be a Pisot number of degree 2. We restrict to (N 1)2+1 d N2 1.
− ≤ ≤ −
Let N tend to infinity and according to (35) consider the values
log N √d
χ(N,d):= | − |, (N 1)2 1 d N2 1.
logN +√d − − ≤ ≤ −
For d = N2 1 we have (N +√d) = (N √d) 1, so in this case ζ = N +√d is an algebraic unit
−
− −
and σ(1,ζ) = σ(1,ζ) = 1. Similarly for d = N2 +1. Conversely, let an arbitrary quadratic irrational
ζ =N +M√d for some M,N,d Z be a unit in Q(√d). Units are known to be precisely the elements
∈