A SHORT NOTE ABOUT DIOPHANTINE APPROXIMATION BY SOME RATIONAL POLYNOMIALS

Abstract

In this short note, we study the Diophantine approximation of real numbers by elements P (t), where 0 < t < 1 and P is a polynomial with coefficient in a set A = {0, q 1 , ..., q N } ⊂ Q with N + 1 ≤ ⌊ 1 t ⌋.

Full text

A SHORT NOTE ABOUT DIOPHANTINE APPROXIMATION
BY SOME RATIONAL POLYNOMIALS
EDOUARD DAVIAUD
Abstract.
In this short note, we study the Diophantine approximation of real
numbers by elements
P(t),
where
0< t < 1
and
P
is a polynomial with coe-
cient in a set
A={0, q1, ..., qN} ⊂ Q
with
N+ 1 ≤ ⌊ 1
t⌋
.
1.
Introduction
In metric Diophantine approximation, one often aims at investigating the Haus-
dor dimension of sets of points by a sequence of of given elements
(xn)n∈N
, chosen
in a meaningful way from the dynamical or algebraic perspective. One way to set
the problem is to look at points
x
that are for innity many elements
xn
at a
distance smaller than
ϕ(n)<< 1,
i.e.
x∈lim sup
n→+∞
B(xn, ϕ(n)).
For instance, the rst historical family of such points where the rational numbers
in
R
and Jarnik and Besicovitch proved the following result.
Theorem 1.1
([8])
.
Consider
δ≥1
and set
Eδ= lim sup
q∈N∗,p∈Z
Bp
q, q−2δ.
A real number
x∈R
is said to be approximable at rate
δ
if
x∈Fδ=Eδ\Sδ′<δ Eδ′.
Then,
dimH(Fδ) = dimH(Eδ) = 1
δ.
Many related question have been investigated:

what about the case where
(xn)n∈N
are the algebraic numbers, root of a
polynomial or natural algebraicly dened numbers?

can we study similar problems in the case of
(xn)n∈N
is the orbit of a
dynamical system ?
Let
N∈N
be an integer and
Q={q1, ..., qN} ⊂ Q
a set of
N
rational numbers.
For
k∈N
we set
(1)
A={0, q1, ..., qN}
and
PA,k =(P(X) =
k
X
i=0
aiXi, ai∈ A)
,
PA=[
k≥0
PA,k.
(2)
1

2 EDOUARD DAVIAUD
In this article, given
t∈(0,1),
we study the case where elements of
(xn)n∈N
are
the elements of
{P(t), P ∈ PA}.
2.
Statement of the result
2.1.
Some notations.
Let us start with some notations
Let
d∈N
. For
x∈Rd
,
r > 0
,
B(x, r)
stands for the closed ball of (
Rd
,
|| ||∞
)
of center
x
and radius
r
. Given a ball
B
,
|B|
stands for the diameter of
B
. For
t≥0
,
δ∈R
and
B=B(x, r)
,
tB
stands for
B(x, tr)
, i.e. the ball with same
center as
B
and radius multiplied by
t
, and the
δ
-contracted ball
Bδ
is dened by
Bδ=B(x, rδ)
.
Given a set
E⊂Rd
,
˚
E
stands for the interior of the
E
,
E
its closure and
∂E =E\˚
E
its boundary. If
E
is a Borel subset of
Rd
, its Borel
σ
-algebra is
denoted by
B(E)
.
Given a topological space
X
, the Borel
σ
-algebra of
X
is denoted
B(X)
and the
space of probability measure on
B(X)
is denoted
M(X).
The
d
-dimensional Lebesgue measure on
(Rd,B(Rd))
is denoted by
Ld
.
For
µ∈ M(Rd)
,
supp(µ) = {x∈[0,1] : ∀r > 0, µ(B(x, r)) >0}
is the topolog-
ical support of
µ
.
Given
E⊂Rd
,
dimH(E)
and
dimP(E)
denote respectively the Hausdor and
the packing dimension of
E
.
2.2.
Main result.
Let
N∈N
be an integer, let
Q={q1, ..., qN} ⊂ Q
a set of
N
rational numbers and
A,PA
as in (1),(2).
Given
t∈(0,1)
and
ϕ:N→R+,
we are interested in investigating the Hausdor
dimension of the set
(3)
WA,t(ϕ) = {x∈R:|x−P(t)| ≤ ϕ(deg(P))
for innitely many
P∈ PA}.
Before stating our result, we need to introduce certain denitions. For
(a1, ..., am)∈
Am,
dene
clt(a0, ..., am−1) = ((a′
0..., a′
m−1)∈ Am:
m−1
X
i=0
aiti=
m−1
X
i=0
a′
iti)
CLt(m) = {clt(a0, ..., am−1),(a0, ..., am−1)∈ Am}.
Dene also,
(4)
s(A, t) = lim sup
m→∞
log #CLt(m)
−log t.
We will show the following result.
Theorem 2.1.
Let
t∈(0,1)
and
ϕ:N→R+
be a mapping such that
ϕ(k)→0,
and set
(5)
sϕ= inf (s:X
k≥1
#CLA,t (k)ϕ(k)s×s(A,t)<+∞).

A SHORT NOTE ABOUT DIOPHANTINE APPROXIMATION BY SOME RATIONAL POLYNOMIALS3
One has
dimHWA,t(ϕ) = min {1, sϕ}sA,t .
In addition, if
t
is transcendental,
sA,t =log(N+1)
−log t.
Note that in Theorem 2.1, the condition
N≤ ⌊1
t⌋ − 1
ensures that
sA,t ≤1.
We
conjecture that, if one replaces
sA,t
by
min {1, sA,t},
then Theorem 2.1 holds for
every
N∈N.
It will be made clear from the proof that the results can be obtained by study-
ing the dynamical coverings associated with some well chosen self-similar IFS. In
the case where
t
is algebraic, then all parameters will be algebraic and one can
use the powerful techniques developed by Hochman in [6]. The case where
t
is
transcendental will be handled using the following very recent result established
by Varju and Rapaport in [9].
Theorem 2.2
([9])
.
Let
S={f1, ..., fm}
be an homogeneous IFS on
R
and
K
its
attractor. Assume that
S
has no-exact overlaps and that, for every
1≤i≤m,
fi(0)
is rational, then
dimHK= min ß1,log m
−log t™.
We also point out that, thanks to a very recent result established by D.J. Feng
and Z. Feng in [5] which extends Theorem 2.2 to homogeneous self-similar IFS with
algebraic translations, one could extend Theorem 2.1 to polynomial with algebraic
coecient instead of just rational.
3.
Proof of Theorem 2.1
For
x∈R,
set
q0= 0
and for every
0≤i≤N, fi(x) = tx +qi.
Dening
S=
{f0, ..., fN}
, we observe that for every
n∈N
and every
(i1, ..., in)∈ {0, ..., N }n,
one has
fi1◦... ◦fin(0) = X
1≤j≤n
qijtj−1.
This implies that
PA,t ={fi1◦... ◦fin(0), n ∈N,(i1, ..., in)∈ {0, ..., N }n}
and
WA,t = lim sup
(i1,...,in)∈Sn≥1{0,...,N}n
B(fi1◦... ◦fin(0), ϕ(|i|)).
The dimension of such sets have been studied in [2] in great generality for homoge-
neous self-similar IFS (such as
S
) and the proof of Theorem 2.1 directly follows by
adapting the techniques developed in this article. Before establishing this result,
we do some recall on the following section about overlapping self-similar IFS.
3.1.
Overlapping IFS and exponential separation condition.
First let us
recall the denition of the packing and Hausdor dimension of a measure
Denition 3.1.
Let
µ∈ M(Rd)
. For
x∈supp(µ)
, the lower and upper local
dimensions of
µ
at
x
are dened as
dimloc(µ, x) = lim inf
r→0+
log(µ(B(x, r)))
log(r)
and
dimloc(µ, x) = lim sup
r→0+
log(µ(B(x, r)))
log(r).

4 EDOUARD DAVIAUD
Then, the lower and upper Hausdor dimensions of
µ
are dened by
(6)
dimH(µ) = ess infµ(dimloc(µ, x))
and
dimP(µ) = ess supµ(dimloc(µ, x))
respectively.
It is known (for more details see [3]) that
dimH(µ) = inf{dimH(E) : E∈ B(Rd), µ(E)>0}
dimP(µ) = inf{dimP(E) : E∈ B(Rd), µ(E)=1}.
When
dimH(µ) = dimP(µ)
, this common value is simply denoted by
dim(µ)
and
µ
is said to be
exact dimensional
.
Let us x
S={f1, ..., fm}
a self-similar IFS on
R
, let
0< c1, ..., cm<1
be the
contraction ratios of
f1, ..., fm
and let
(p1, ..., pm)
be a probability vector. Huntchin-
son's Theorem [7] implies that there exists a unique non empty compact set and
a a unique probability measure
µ∈ M(R)
satisfying that
(7)
®K=S1≤i≤mfi(K)
µ(·) = P1≤i≤mpiµ(f−1
i(·)).
Due to a result of Feng and Hu [4],
µ
is known to be exact-dimensional. Let us
write
Λ = {1, ..., m},Λ∗=Sn≥1Λn
and Given
i= (i1, ..., in)∈Λn
, we introduce
some notations:

Given numbers
α1, ..., αm,
we write
αi=αi1×... ×αin
and
fi=fi1◦... ◦fin,

we denote
cl(i) = ¶j∈Λn:fj=fi©
and
cln={cl(i), i ∈Λn},

we also set
epi=X
j∈cl(i)
pj.
In the rest of the paper, when there is no ambiguity, we will not distinguish between
a word
i
and its class
cl(i).
Denition 3.2.
Fix
n∈N
and set
Hn(µ) = X
i∈clnepilog epi.
The Garcia entropy and of
µ
is dened as
HG(µ) = lim
n→+∞
1
nHn(µ).
In addition, denoting for every
n∈N, sn
the solution to
X
i∈cln
csn
i= 1,
the Garcia dimension of
S
is dened as
dimG(S) = lim inf
n→+∞sn.

A SHORT NOTE ABOUT DIOPHANTINE APPROXIMATION BY SOME RATIONAL POLYNOMIALS5
Let us also add that, when the IFS is homogeneous, it is easily seen that
(sn)n∈N
converges. In addition, if the IFS has no exact-overlaps (i.e. for every
i=j,
fi=fj
), then calling the similarity dimension of
Sdim(S)
, i.e.
dim(S)
is the
solution to
X
1≤i≤m
cdim(S)
i= 1,
one has
dimG(S) = dim(S).
In [6], it is established that the exact overlap conjecture holds provided that
S
satises the so-called exponential separation condition, which forbids exact-
overlaps. In many articles, it was noticed that the actual proof can be adapted
to the exact-overlaps case by assuming exponential separation for the maps cor-
responding to dierent classes in
cln
. For the seek of completeness, the precise
statements and the proof of this result are included in the homogeneous case
(which sucient for the purpose of this article).
Theorem 3.1.
Let
S={f1, ..., fm}
be an homogeneous self-similar IFS on
R
of
contraction ratio
t∈(0,1)
and
K
its attractor. Fix
x∈K
dene
(8)
∆n= min ¶|fi(x)−fj(x)|, i =j∈cln©.
Assume that there exists
0< c < 1
such that, for every large enough
n∈N,
one
has
∆n≥cn,
then:
(1)
for every probability vector
P= (p1, ..., pm)
the self-similar measure
µ
as-
sociated with
P
satises
dim(µ) = min ß1,HG(µ)
−log t™.
(2)
one has
dimHK= min {1,dimG(S)}.
Proof.
First, by changing the coordinates, we may assume that
x= 0
, which we
do.
We rst establish item
(1) :
For
n∈N,
denote
Dn
the set of dyadic interval of generation
n
, write
n′=
⌊nlog t
−log 2 ⌋
and set
ν(n)=X
i∈Λn
piδfi(0)
and observe that
ν(n)=X
i∈clnepiδfi(0).
Given
E
an at most countable Borel partition of
R
, set
H(µ, E) = X
E∈E
−µ(E) log µ(E).
The following theorem, established in [6] is the key of our proof.
Theorem 3.2.
If
dim(µ)<1,
then for every
q∈N,
(9)
lim
n→+∞
1
n′H(ν(n),Dqn′)−H(ν(n),Dn′)= 0.

6 EDOUARD DAVIAUD
If
dim µ= 1,
then item
(1)
holds. Assume that
dim(µ)<1,
x
q
so large that
2qlog t
log 2 < c
and x
ε > 0
and
n0
large enough so that for every
n≥n0,
(1 −ε)HG(µ)
−log t≤Pi∈cln−epilog epi
−nlog t≤(1 + ε)HG(µ)
−log t.
Note that, for every
n, 2−qn′< cn
, so, recalling that
∆n≥cn,
no elements of
Dqn′
can contain two atoms of
ν(n).
This implies that
H(ν(n),Dqn′) = X
i∈cln
−epilog epi.
Since
limn→01
n′H(νn,Dn′) = dim(µ)
, by (9), one has
(1 −ε)HG(µ)
−log t≤dim(µ)≤(1 + ε)HG(µ)
−log t.
Letting
ε→0
yields item (1).
Let us prove that item (2) holds. Notice rst that for every
n∈N,{fi(K), i ∈cln}
covers
K
. Fix
ε > 0
and let
n0
be so large that for every
n≥n0,
dimG(S)−ε≤sn≤dimG(S) + ε.
One has
X
i∈cln
|fi(K)|dimG(S)+ε≤ |K|dimG(S)+εX
i∈cln
tsn=|K|dimG(S)+ε.
This proves that
HdimG(S)+ε<+∞
, so, letting
ε→0,
since
dimHK≤1,
one gets
dimHK≤min {1,dimG(S)}.
Moreover, let
Sn={fi, i ∈cln}
and, for
i∈cln,
set
pi=tsn
. Let
µn
be the
self-similar measure associated with
(pi)i∈cln
and
Sn,
one has
HG(µn)≥dimG(S)−ε,
so applying item (1), one obtains
dimHK≥dim(µn)≥min {1,dimG(S)−ε}.
Letting
ε→0
nishes the proof.
□
3.2.
Recall and dynamical covering associated with homogeneous self-
similar IFS.
Let
S
be an homogeneous self-similar IFS. The following result is
established in [2].
Theorem 3.3
([2])
.
Let
S
be an homogeneous self-similar IFS. Let
ϕ:N→
(0,+∞)
be a such that
limn→+∞ϕ(n) = 0
and set
(10)
sϕ= inf 

s≥0 : X
k≥0X
i∈Λk
ϕ(k)sdim(S)<+∞

.
Assume that
dimHK= dim(S)
. Then
dimHW(x0, ϕ) := lim sup
i∈Λ∗
B(fi(x0), ϕ(|i|)) = min {1, sϕ}dim(S).
Let us start by showing that Theorem 3.3 actually holds by replacing
dim(S)
by
dimG(S)
in the statement, i.e., that the following statement holds.

A SHORT NOTE ABOUT DIOPHANTINE APPROXIMATION BY SOME RATIONAL POLYNOMIALS7
Theorem 3.4.
Let
S
be an homogeneous self-similar IFS. Let
ϕ:N→(0,+∞)
be a such that
limn→+∞ϕ(n) = 0
and set
(11)
sϕ= inf 

s≥0 : X
k≥0X
i∈Λk
ϕ(k)sdimG(S)<+∞

.
Assume that
dimHK= dimG(S)
. Then
dimHW(x0, ϕ) := lim sup
i∈Λ∗
B(fi(x0), ϕ(|i|)) = min {1, sϕ}dimG(S).
Lemma 3.5.
Let
ψ:N→R+
be a mapping such that
X
n≥1X
i∈cln
ψ(n)dimG(S)= +∞,
then, for every
ε > 0
, there exists an innity of integers
(nk)k∈N
such that
ψ(nk)≥
t(1+ε)nk.
Proof.
Assume that it is not the case: there exists
ε > 0
and
N∈N
such that for
every
n > N, ψ(n)< t(1+ε)n
and
dimG(S)≥sn
1+ε+ε2.
In this case, recalling that
for every
n∈N,Pi∈clntnsn= 1,
X
n≥NX
i∈cln
ψ(n)dimG(S)≤X
n≥NX
i∈cln
tnsn+n(1+ε)ε2
=X
n≥N
tn(1+ε)ε2X
i∈cln
tnsn<+∞,
which is a contradiction.
□
This lemma implies in particular that for every
ε > 0,
there exists an innity of
integers
(nk)k∈N
for every
i∈cnk
, writing
1−ε=1
1+ε′,
ψ(nk)1−ε≥ |fi(K)|.
Since for every
k∈N
and every
x0∈K,
K⊂[
i∈cnk
B(fi(x0),|fi(K)|),
one obtains the following corollary.
Corollary 3.6.
Let
ψ:N→R+
be a mapping such that
X
n≥1X
i∈cln
ψ(n)dimG(S)= +∞,
then, for every
x0∈K
and every
ε > 0
,
K= lim sup
i∈Λ∗
B(fi(x0), ψ(n)1−ε).
Let us also recall the two following results.
Theorem 3.7
([1])
.
Let
S
be a self-similar IFS and let
µ
be a self-similar measure
associated with
S
.
Let
(Bn)n∈N
be a sequence of closed balls centered on
supp(µ)
with
limn→+∞|Bn|=
0
. If
µ(lim sup
n→+∞
Bn)=1,

8 EDOUARD DAVIAUD
then, for every
δ≥1,
(12)
dimH(lim sup
n→+∞
Bδ
n)≥dim(µ)
δ.
Proposition 3.8
([2])
.
Let
S
be a self-similar IFS, let
K
be its attractor and
ε0>0
. There exists an IFS
Sε0
and a self-similar measure
µε0
(associated with
Sε0
) such that
supp(µε0) = K
and
dimH(µε0)≥dimHK−ε0.
We now prove Theorem 3.3.
Recall (10) and assume rst that
sϕ≥1.
Then by Lemma 3.5, for every
ε > 0,
K= lim sup
i∈Λ∗
B(fi(x0), ϕ(n)1−ε).
Let
µ∈ M(Rd)
be given by Proposition 3.8, one has
µlim sup
i∈Λ∗
B(fi(x0), ϕ(n)1−ε)= 1,
which, by Theorem 3.7, implies that, writing
δε=1
1−ε
dimHW(x0, ϕ)≥dimHK−ε
1
1−ε
.
Since this holds for every
ε > 0,
one has
dimHW(x0, ϕ)≥dimHK,
hence
dimHW(x0, ϕ) = dimHK= min {1, sϕ}dimG(S).
We now assume that
sϕ<1.
We rst prove that
dimHW(x0, ϕ)≤sϕdimG(S).
Let
ε > 0.
By denition of
sϕ
(see (10)),
X
n≥1X
i∈cln
ϕ(n)(sϕ+ε) dimG(S)<+∞,
which yields
H(sϕ+ε) dimG(S)(W(x0, ϕ)) = 0,
since this holds for every
ε > 0,
dimHW(x0, ϕ)≤sϕdimG(S).
Let us show that
dimHW(x0, ϕ)≥sϕdimG(S).
Fix again
ε > 0.
Since
X
n≥1X
i∈cln
ϕ(n)(sϕ−ε) dimG(S)= +∞,
by Lemma 3.5, one has
K= lim sup
i∈Λ∗
B(fi(x0), ψ(n)(sϕ−ε)(1−ε)).
Let
µ∈ M(Rd)
be given by Proposition 3.8. One has
µW(x0, ϕ(sϕ−ε)(1−ε))= 1.

A SHORT NOTE ABOUT DIOPHANTINE APPROXIMATION BY SOME RATIONAL POLYNOMIALS9
Writing
δε=1
(sϕ−ε)(1−ε)
and applying Theorem 3.7, one gets
dimHW(x0, ϕ)≥dim(S)
δε
= (sϕ−ε)(1 −ε) dimG(S).
Letting
ε→0,
yields
dimHW(x0, ϕ)≥sϕdimG(S),
which concludes the proof.
3.3.
Application to the proof of Theorem 2.1.
Consider as in the beginning
of Section 3,
f0(x) = tx +qi
,
S={f0, ..., fN}
and call
K
its attractor. Notice that
0∈K
and that
sA,t = dimG(S).
Recall also that
WA,t = lim sup
(i1,...,in)∈Sn≥1{0,...,N}n
B(fi1◦... ◦fin(0), ϕ(|i|)).
In addition, when
t
is algebraic, then by [6, Lemma 5.10],
S
there exists
0<c<1
such that, for every large enough
n∈N,
one has
∆n≥cn,
where
∆n
is dened as in (8). So applying Theorem 3.1, one has
dimHK= dimG(S)
and applying Theorem 3.4 yields item
(1).
In addition if
t
is not algebraic,
S
has no exact overlaps and, calling
K
its
attractor, by Theorem 2.2, one has
dimHK= dim(S) = sA,t =log(N+ 1)
−log t.
Applying Theorem 3.3 yields the result.
References
[1] E. Daviaud. A dimensional mass transference principle for borel probability measures and
applications. Submitted, arXiv:2204.01302, 2022.
[2] E. Daviaud. Dynamical coverings for
C1
weakly conformal IFS with overlaps. Submitted ,
arXiv:2207.08458, 2023.
[3] K. Falconer. Fractal geometry. John Wiley & Sons, Inc., Hoboken, NJ, second edition, 2003.
Mathematical foundations and applications.
[4] D. Feng and H. Hu. Dimension theory of iterated function systems. Comm. Pure Appl. Math.,
62:14351500, 2009.
[5] D.J. Feng and Z. Feng. Ddimension of homogeneous iterated function systems with algebraic
translations. in preparation, 2023.
[6] M. Hochman. On self-similar sets with overlaps and inverse theorems for entropy in
Rd
. To
appear in Memoires of the AMS, 2022.
[7] J.E. Hutchinson. Fractals and self similarity. Indiana Univ. Math. J., 30:713747, 1981.
[8] V. Jarnik. Diophantischen approximationen und Hausdorsches mass. Mat. Sbornik, 36:371
381, 1929.
[9] A. Rapaport. Self-similar measures associated to a homogeneous system of three maps. Duke,
2023.