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arXiv:1208.2089v1 [math.NT] 10 Aug 2012
INTRINSIC APPROXIMATION FOR FRACTALS DEFINED BY
RATIONAL ITERATED FUNCTION SYSTEMS - MAHLER’S
RESEARCH SUGGESTION
LIOR FISHMAN AND DAVID SIMMONS
Abstract. Following K. Mahler’s suggestion for further research on intrinsic approxi-
mation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets
of rational iterated function systems. We further investigate the behavior of these ap-
proximation functions under random translations. We connect the information regarding
the distribution of rationals on the limit set encoded in the system to the distribution of
rationals in reduced form by proving a Khinchin type theorem. Finally, using a result of
S. Ramanujan, we prove a theorem motivating a conjecture regarding the distribution of
rationals in reduced form on the Cantor ternary set.
1. Introduction
In 1984, K. Mahler published a paper entitled “Some suggestions for further research”
[9], in which he writes the following moving statement: “At the age of 80 I cannot expect
to do much more mathematics. I may however state a number of questions where perhaps
further research might lead to interesting results”. One of these questions was regarding
intrinsic and extrinsic approximation on the Cantor set. In Mahler’s words, “How close
can irrational elements of Cantor’s set be approximated by rational numbers
(1) In Cantor’s set, and
(2) By rational numbers not in Cantor’s set?”1
In contrast to intrinsic approximation on the Cantor set in particular and on fractals in
general, the Diophantine approximation theory of the real line is classical, extensive, and
essentially complete as far as characterizing how well real numbers can be approximated
by rationals ([11] is a standard reference). The basic result on approximability of all reals
is
Theorem (Dirichlet’s Approximation Theorem).For each x∈Rand for any Q∈Nthere
exists p/q ∈Qwith 1≤q≤Q, such that
x−p/q<1
qQ .
Corollary. For every irrational x∈R,
x−p/q<1
q2for infinitely many p/q ∈Q.
1Our paper is mainly concerned with the first question, but see discussion at the end of the introduction
regarding the second.
1
2 L. FISHMAN AND D. SIMMONS
The optimality of this approximation function (up to a multiplicative constant) is demon-
strated by the existence of badly-approximable numbers, i.e., the set of reals xsuch that
for some c(x)>0x−p/q>c(x)
q2for all p/q ∈Q.
The well known fact that the set of very well approximable numbers, i.e., the set of all
reals xsatisfying for some positive ǫ(x)
x−p/q<1
q2+ǫ(x)
for infinitely many rationals p/q is null, demonstrates that this approximation function can-
not be improved for almost all irrationals. We remark that the subject of approximating
points on fractals by rationals has been extensively studied in recent years; see for example
[3], [6], and [7] for badly approximable numbers and [5] and [12] for very well approximable
numbers. In [2], elements of the middle third Cantor set with any prescribed irrationality
exponent were explicitly constructed.
Recently in [1], R. Broderick, A. Reich, and the first named author made what could be
considered as a first step towards answering Mahler’s question:
Proposition ([1] Corollary 2.2).Let Cbe the ternary Cantor set and d= dim C. Then
for all x∈C, there exist infinitely many solutions p∈N,q∈N,p/q ∈Cto
x−p/q<1
q(log3q)1/d .
The proof of the above proposition crucially depends on the ×3 invariance of the mid-
dle third Cantor set, and a similar more general result holds for any ×d-invariant totally
disconnected Cantor-like set.
The main motivation of this paper is not only to generalize the results in [1] but to try
and provide a better understanding of Diophantine approximation on fractals in general. In
Section 2 we generalize Proposition 1 namely by removing the ×dconstraint. We consider
the following setup:
Definition 1.1. Let Jbe a subset of Rand let ψ:N→(0,∞) be any function. A point
x∈Jis said to be intrinsically approximable with respect to ψif there exist infinitely many
rationals p/q ∈Q∩Jsuch that
|x−p/q| ≤ ψ(q)
q.
xis said to be badly intrinsically approximable with respect to ψif there exists ε > 0 such
that xis not intrinsically approximable with respect to the function εψ. Otherwise, xis
said to be intrinsically well approximable with respect to ψ.
Definition 1.2. Let Ebe a finite set. A rational iterated function system is an iterated
function system (ua)a∈Esatisfying the open set condition2and acting on Rconsisting of
2The open set condition is a standard requirement. See [4] for a thorough discussion.
INTRINSIC APPROXIMATION FOR FRACTALS 3
contracting similarities preserving Q. In other words, for each a∈Ewe have
(1.1) ua(x) = pa
qa
x+ra
qa
where pa
qa
,ra
qa
∈Q.
Theorem 2.1. Suppose that (ua)ais a rational IFS and let Jbe the limit set of this IFS.
Let
γ:= max
a∈E
ln |pa|
ln(qa),
where pa, qaare as in (1.1). There exists K < ∞such that for each x∈Jand for each
Q≥qmax := maxaqathere exists p/q ∈Q∩Jwith q≤Qsuch that
|x−p/q| ≤ Kqγ−1ln(Q)−1/δ.
In particular, if xis irrational then xis intrinsically approximable with respect to the
function
ψ(q) := Kqγln(q)−1/δ .
Notice that the Dirichlet-type theorems in [1] are immediate consequences of Theorem
2.1 as γ= 0 whenever pa=±1 for all a∈E.
In Section 3 we consider translations of ×d-invariant limit sets Jof a rational IFS of
Hausdorff dimension δ. A ×d-invariant limit set is the set of all points x∈[0,1] such that
the digits of the base dexpansion of xare contained in some fixed set E⊆ {0,...,d−1};
for example the ternary Cantor set is ×3-invariant. In this case, we wish to investigate how
“generic” the approximation function in Theorem 2.1 is with respect to random translations.
For each x∈Rlet Jx=J−xand consider the set Q∩Jx. Note that Q∩Jxis nonempty
if and only if x=y−p/q for some y∈Jand for some p/q ∈Q. More formally, the set of x
for which Q∩Jx6=∅is the set Sp/q∈QJp/q. In particular, this set has Hausdorff dimension
δbut Lebesgue measure zero. Thus if a translate of Jis picked “at random”, i.e. according
to Lebesgue measure, then it will contain no rationals. Clearly, we have so far not used the
fractal nature of J, only the fact that it is a Lebesgue null set. An immediate question is
what happens if Q∩Jxis nonempty:
Lemma 3.1. Suppose that Jis a ×d-invariant limit set and suppose that x∈Ris such
that Q∩Jx6=∅. Then Q∩Jxis dense in Jx. Furthermore, there exists K < ∞such that
every point y∈Jxis intrinsically approximable with respect to the constant function
ψ(q) = K.
A natural question is whether the approximation function is optimal. The simplest case
is the following:
i) dis prime
ii) The set Eof allowable digits contains no two adjacent integers
iii) Every rational in Jxis of the form p/dnfor some p, n ∈N.
This case is similar to the case of approximating elements of the Cantor set by only the
endpoints, which was studied by Levesley, Salp, and Velani, who proved a Khinchin-type
theorem and a Jarnik-Besicovitch-type theorem (see [8]). By translating their results into
our setting we can prove the following theorem:
4 L. FISHMAN AND D. SIMMONS
Theorem 3.6. Let Jand xsatisfy Case 0 and fix any function ψ:N→(0,∞). Let
fbe a dimension function such that t7→ t−δf(t)is monotonic. If we denote the set of
ψ-intrinsically well approximable points by WAψ,int, then
HDf(WAψ,int) = (0if P∞
n=1 f(ψ(dn)/dn)(dn)δ<∞
HDf(J)if P∞
n=1 f(ψ(dn)/dn)(dn)δ=∞.
Corollary 3.7. If ψ(q) = ln(q)−1/δ , then almost every point is intrinsically well approx-
imable with respect to ψ; if ψ(q) = ln(q)−(1+ε)/δ, then almost every point is badly intrinsi-
cally approximable with respect to ψ.
Additionally, we prove the following theorem demonstrating the optimality of the Dirich-
let approximation function ψ(q) = 1:
Theorem 3.5. Let Jand xsatisfy (i)-(iii) and let δbe the Hausdorff dimension of J.
Then the set of numbers y∈Jxwhich are badly intrinsically approximable with respect to
the approximation function ψ(q) = Kis of Hausdorff dimension δ.
In fact, the case in which (i)-(iii) hold is the only case in which we are able to say anything
about the optimality of the approximation function. We remark that (i)-(ii) are reasonable
and easy-to-check assumptions that are satisfied e.g. for the standard ternary Cantor set.
Finally, we prove that condition (iii) is generic:
Theorem 3.2. Suppose that Jis a ×d-invariant limit set satisfying (i)-(ii). The set S
consisting of all x∈Sp/q∈QJp/q for which (iii) does NOT hold is small with respect to both
measure and category, i.e. Hδ(S) = 0 and S∩Jis meager in J.
In Sections 4 and 5 we discuss the question of whether the approximation function of
Theorem 2.1 is optimal. The starting point is the observation that the method of Theorem
2.1 produces only rational numbers of a particular form. Specifically, if we let π:EN→J
be the coding map (defined precisely in Section 2), then Theorem 2.1 produces rationals of
the form π(ω), where ω∈ENis an eventually periodic word.
Secondly, when Theorem 2.1 does produce a rational number then it does not produce
it in reduced form. For example, the fraction 1/4 in the ternary Cantor set Cwould be
represented as 2/8. Consequently, we will call the number 8 the intrinsic denominator of
1/4 with respect to the fractal C(defined precisely in Section 4). Note that a rational
can only have an intrinsic denominator if it has a preimage which is an eventually periodic
word. We denote the intrinsic denominator of p/q given by the IFS by qint, whereas the
denominator of p/q in reduced form will be denoted qred. (Following our above example,
qint = 8 while qred = 4). It is easily observed that for every p/q ∈Jwe have qred ↿qint.
As a result of this analysis, the question of whether Theorem 2.1 is optimal can now be
split into three sub-questions:
(i) Does every rational in Jhave a preimage under πwhich is eventually periodic? In
other words, are the rationals in Jthat have intrinsic denominators the only ones that
exist?
(ii) If p/q is a rational in Jthat has an intrinsic denominator, what is the ratio between
its intrinsic denominator qint and its reduced denominator qred?
INTRINSIC APPROXIMATION FOR FRACTALS 5
(iii) Is the approximation function (2.2) optimal if we only consider rationals in Jwhich
come from periodic words, and if we consider the intrinsic denominator to be the true
denominator of the rational?
In Section 4 we will consider questions (i) and (iii), and in Section 5 we will consider
question (ii). In each case we have only partial results. In appears that all three questions
are hard when considered in full generality, although it seems (ii) is the hardest.
Question (i) is the easiest to deal with. In the case of the ternary Cantor set (or more
generally of a ×d-invariant set), the answer is already well-known. The fact that every
rational in Jhas a preimage under πwhich is eventually periodic is merely a restatement
of the fact that every rational number has an eventually periodic base dexpansion. We
slightly generalize this result with the following lemma:
Lemma 4.2. Suppose that pa=±1for all a∈E, where paare given by (1.1). Then every
rational in Jis the image of an eventually periodic word (and therefore has an intrinsic
denominator).
We next consider question (iii):
Definition 4.7. Let ψ: (0,∞)→(0,∞)be a nonincreasing function. A point x∈Jis
said to be badly symbolically approximable with respect to ψif there exists ε > 0such that
for all p/q ∈Q∩Jwe have
|x−p/q| ≥ εψ(qint)
qint
.
Otherwise, xis said to be symbolically well approximable with respect to ψ.
It thus follows that badly intrinsically approximable implies badly symbolically approx-
imable, but not vice-versa.
Rather than attempting to demonstrate the existence of numbers which are badly sym-
bolically approximable with respect to the Dirichlet function (2.2), we instead prove a
Khinchin-type theorem. Our motivation for this is that it seems less likely that the in-
trinsic denominator differs greatly from the denominator in reduced form for the rational
approximations of almost every point, than that it differs greatly for the approximants of
a single point.
An immediate corollary of Section 4’s main theorem (Theorem 4.12) is the following:
Corollary 4.13. Let Cbe the ternary Cantor set and µthe Hausdorff measure in the Can-
tor’s set dimension restricted to C. Then for µ-almost every x∈J,xis badly symbolically
approximable with respect to ψ(q) = ln(q)−(2/δ+ε)and is symbolically well approximable with
respect to ψ(q) = ln(q)−2/δ.
In Section 5, we restrict ourself to the case where the limit set is the Cantor ternary set
C. We begin by recalling the following conjecture from [1]:
Conjecture 5.1 ([1] Conjecture 3.3).If
Sn:= {p/q ∈J: gcd(p, q) = 1,3n−1≤q < 3n}
then for all ε1>0we have
#(Sn)∈O(2n(1+ε1)).
6 L. FISHMAN AND D. SIMMONS
This conjecture is immediately relevant to intrinsic approximation as it implies [[1] Corol-
lary 3.4] that µ(VWAJ) = 0, where
VWAJ:= {x∈J:∃ε > 0∃∞p/q ∈J|x−p/q| ≤ q−(1+ε)}.
We cannot prove Conjecture 5.1 at this time, but we will reduce it to a simpler conjecture
which a heuristic argument suggests is true. Suppose that p/q is a rational number. The
period of p/q is the period of the ternary expansion of p/q, and will be denoted P(p/q).
The first step we make is proving the following theorem (using a result of Ramanujan
[10] concerning the number-of-divisors function):
Theorem 5.3. For every K < ∞, if
S(K)
n:= {p/q ∈J: gcd(p, q) = 1,3n−1≤q < 3n,and P(p/q)≤Kln(q)}
then for all ε1>0we have
#(S(K)
n)∈O(2n(1+ε1)).
Next, we provide a heuristic argument to support the following conjecture:
Conjecture 5.6. For all K > 2/ln(3/2), we have S(K)
n=Snfor all nsufficiently large.
In particular
#(Sn\S(K)
n)∈o(1).
The following is a corollary of Theorem 5.3:
Corollary 5.4. Conjecture 5.6 implies Conjecture 5.1, implying µ(VWAJ) = 0.
Finally, let us note that Conjecture 5.1 also has relevance to Mahler’s second question
regarding extrinsic approximation. Indeed, if xis any point which is badly intrinsically
approximable with respect to the function
ψ(q) = q−1+ε,
for some ε > 0, then xis extrinsically approximable with respect to Dirichlet’s function
ψ(q) = q−1. To see this, note that by Dirichlet’s Theorem there is a sequence pn/qn−→
nx
such that |x−pn/qn| ≤ 1/q2
n, but since xis badly intrinsically approximable with respect
to ψ(q) = q−1+ε, it follows that only finitely many of these approximations can be intrinsic.
Thus if Conjecture 5.1 is correct, then almost every point on the Cantor set is extrinsically
approximable with respect to the function ψ(q) = q−1.
Acknowledgements. Both authors would like to thank Y. Bugeaud and M. Urba´nski
for helpful suggestions and comments.
INTRINSIC APPROXIMATION FOR FRACTALS 7
2. A Dirichlet-type theorem for fractals
We consider a finite set (alphabet) Eand denote by Erthe set of all words of length r
formed using this alphabet and by E∗the set of all words, finite or infinite, formed using
this alphabet. If ω∈E∗, then we denote subwords of ωby
ωn+r
n+1 := (ωn+i)r
i=1 ∈Er.
We denote the concatenation of ωand τby ω∗τ. Furthermore, we define the shift map
σ:EN→EN
by σ(ω) = ω∞
2= (ωi+1)i∈N. If ω∈Enis a finite word then we define
uω(x) := uω1◦...◦uωn(x).
We define the map π:EN→Rby
π(ω) := lim
n→∞ uωn
1(0)
and we define the limit set Jto be the image of this map. Let δbe the Hausdorff dimension
of J, and let µbe the δ-dimensional Hausdorff measure restricted to J, normalized to be a
probability measure.
Theorem 2.1 (Dirichlet for fractals).Suppose that (ua)ais a rational IFS and let Jbe the
limit set of this IFS. Let
(2.1) γ:= max
a∈E
ln |pa|
ln(qa),
where pa, qaare as in (1.1). There exists K < ∞such that for each x∈Jand for each
Q≥qmax := maxaqathere exists p/q ∈Q∩Jwith q≤Qsuch that
|x−p/q| ≤ Kqγ−1ln(Q)−1/δ.
In particular, if xis irrational then xis intrinsically approximable with respect to the
function
(2.2) ψ(q) := Kqγln(q)−1/δ .
Proof. Recall that the measure µ:= Hδ↿J/Hδ(J) is Ahlfors δ-regular on J, i.e.
µ(B(x, r)) ≍rδ
where x∈Jand 0 < r ≤1. Thus if (xn)N−1
n=0 is an r-separated sequence in J, i.e. if
d(xn, xm)≥rfor all n6=m, then the balls (B(xn, r/2))N−1
n=0 are disjoint and so
1 = µ(J)≥
N−1
X
n=0
µ(B(xn, r/2)) ≍
N−1
X
n=0
(r/2)δ≍N rδ.
Thus there exists C1<∞depending only on Jsuch that Nrδ≤C1. Taking the contra-
positive gives
8 L. FISHMAN AND D. SIMMONS
Lemma 2.2 (Fractal pigeonhole principle).If (xn)N
n=0 is any finite sequence in J, then
there exist distinct integers 0≤n, m ≤Nsuch that
|xn−xn+m| ≤ rN:= (N/C1)−1/δ .
Now suppose we have x∈Jand Q≥qmax; fix N∈Nto be determined. Let ω∈ENbe
a preimage of xunder π. We consider the iterates of ωunder the shift map σ. We apply
the fractal pigeonhole principle to the sequence (π◦σn(ω))N
n=0 to conclude that there exist
two integers 0 ≤n < n +m≤Nsuch that if y=π◦σn(ω) and z=π◦σn+m(ω), then
|y−z| ≤ rN.
Let u(1) =uωn
1and let u(2) =uωn+m
n+1 . Then x=u(1)(y), and y=u(2) (z). Let
p(1) =pω1···pωnp(2) =pωn+1 ···pωn+m
q(1) =qω1···qωnq(2) =qωn+1 ···qωn+m
r(1) =
n
X
i=1
pω1···pωi−1rωiqωi+1 ···qωnr(2) =
m
X
i=1
pωn+1 ···pωn+i−1rωn+iqωn+i+1 ···qωn+m
so that
u(i)(t) = p(i)
q(i)
t+r(i)
q(i)
.
The unique fixed point F2of the contraction u(2) is given by the equation
F2=p(2)
q(2)
F2+r(2)
q(2)
and after solving for F2
F2=r(2)
q(2) −p(2)
.
In particular, F2∈Q. Let
(2.3) p/q =u(1)(F2) = p(1)
q(1)
r(2)
q(2) −p(2)
+r(1)
q(1)
.
Here, we mean that p∈Z,q∈Nare the result of adding the fractions in the usual
way, without reducing. In particular, q=q(1)(q(2) −p(2))≤q(1)q(2). Note that p/q =
π(ωn
1∗[ωn+m
n+1 ]∞)∈J.
We next want to bound the distance between xand p/q. For convenience of notation let
λ(i)=|p(i)|/q(i)be the contraction ration of u(i). Now
|y−F2|=λ(2)|z−F2| ≤ λ(2) |y−F2|+λ(2)|z−y|;
solving for |y−F2|gives
|y−F2| ≤ λ(2)
1−λ(2)
|z−y|.
Applying u(1) gives
|x−p/q| ≤ λ(1)λ(2)
1−λ(2)
|z−y| ≤ λ(1) λ(2)
1−λ(2)
rN.
Now λ(2) ≤maxaλa<1; if we let C2= 1/(1 −maxaλa) then
|x−p/q| ≤ C2λ(1) λ(2)rN
INTRINSIC APPROXIMATION FOR FRACTALS 9
and on the other hand
q≤q(1)q(2).
Expanding and taking logarithms
ln |x−p/q| ≤ ln(C2) + ln(rN) +
n+m
X
k=1
ln(λωk)
ln(q)≤
n+m
X
k=1
ln(qωk).
Now for every i= 1,...,m we have
ln(λi)≤(γ−1) ln(qi)
where γis as in (2.1). Thus
ln |x−p/q| ≤ ln(C2) + ln(rN) + (γ−1)
n+m
X
k=1
ln(qωk)
≤ln(C2) + ln(rN) + (γ−1) ln(q)
and rearranging gives
|x−p/q| ≤ C2rNqγ−1.
Finally, recalling that qmax := maxaqa, we have
q≤qn+m
max ≤qN
max
and so letting N=⌊logqmax (Q)⌋gives q≤Q. Now since Q≥qmax, we have
rN≤(⌊logqmax (Q)⌋/C1)−1/δ ≤C3ln(Q)−1/δ
for some C3sufficiently large. Letting K=C2C3completes the proof.
3. Random translations
Fix d∈Nand E⊆ {0, . . . , d −1}satisfying 1 <#(E)< d, and let
J={x∈[0,1] : the digits of the base dexpansion of xare in E}.
Such a set Jis called a ×d-invariant set.
For each x∈Rlet Jx=J−xand consider the set Q∩Jx. As observed in the Introduction,
the set of xfor which Q∩Jx6=∅is the set Sp/q∈QJp/q , which has Hausdorff dimension δ.
In this section, we consider the case where xlies in this set, so that Q∩Jx6=∅.
Lemma 3.1. For all x∈R, if Q∩Jxis nonempty, then Q∩Jxis dense in Jx. Furthermore,
there exists K < ∞such that every point y∈Jxis intrinsically approximable with respect
to the constant function
(3.1) ψ(q) = K.
10 L. FISHMAN AND D. SIMMONS
Proof. Fix y∈Jxand p/q ∈Q∩Jx. For each n∈N, let znbe the first ndigits of x+y∈J
spliced with the last digits of x+p/q ∈J. We observe that zn∈J, since the digits of the
base dexpansion of znlie in E. Then rn:= zn−x−p/q is a multiple of 1/dn. In particular,
p/q +rn=zn−x∈Q∩Jand
denom(p/q +rn)≤qdn.
Now
|y−(p/q +rn)|=|zn−(x+y)| ≤ d−n
since zagrees with (x+y) up to the first ndigits. Thus
|y−(p/q +rn)| ≤ q
denom(p/q +rn),
proving the lemma with K=q.
We now discuss optimality of the approximation function (3.1). The simplest case is the
following:
i) dis prime
ii) The set Eof allowable digits contains no two adjacent integers
iii) Every rational in Jxis of the form p/dnfor some p, n ∈N.
We will call this case Case 0.
In Case 0, the approximation function (3.1) is optimal in the following two senses:
•The set of badly intrinsically approximable numbers is of full Hausdorff dimension
(Theorem 3.5)
•A Khinchin-type theorem holds (Theorem 3.6)
In fact, Case 0 seems to be the only case in which it is easy to say anything about the
optimality of (3.1). Nevertheless, we prove the following:
Theorem 3.2. Suppose that Jis a ×d-invariant limit set satisfying (i)-(ii). The set S
consisting of all x∈Sp/q∈QJp/q for which (iii) does not hold is small with respect to both
measure and category, i.e. Hδ(S) = 0 and S∩Jis meager in J.
Proof. It suffices to show the following:
Claim 3.3. If x∈Jhas a base dexpansion x=P∞
i=1 aid−iwhich contains every finite
word as a substring, then Case 0 holds.
By way of contradiction, suppose that xhas the above property, but there exists a
rational p/q ∈Q∩Jxwhose denominator is not a power of d. Without loss of generality
assume p/q > 0; the case p/q < 0 is similar. Let p/q =P∞
i=1 bid−ibe the base dexpansion
of p/q, and let x+p/q =P∞
i=1 cid−i∈J. Then ci=ai+bior ci=ai+bi+ 1, mod d,
depending on whether there is a carry from the lower level terms. Write
p/q =.b1. . . bn+mbn+1 ...
for some n, m ∈N. Let k=bn+m. By a counting argument there exists a∈Esuch that
either a+k /∈E, or a+k+ 1 /∈E. Without loss of generality suppose that a+k /∈E.
Let τ= (a0m)n+m. (If a+k+ 1 /∈Ewe take τ= (a(d−1)m)n+m.) By assumption τis a
INTRINSIC APPROXIMATION FOR FRACTALS 11
substring of (ai)i. Thus we can find a0mas a substring of (ai)isuch that the acorresponds
to an occurence of kin p/q, i.e. there exists i∈Nsuch that
an+mi =a
an+mi+j= 0
for all j= 1,...,m. Now since qis not a power of d, we have bn+j6=d−1 for some
j= 1,...,m. Thus the zeros (an+mi+j)m
j=1 are sufficient to ensure that there is not a carry
in the (n+mi)th place, implying cn+mi =an+mi +bn+mi =a+k /∈E, a contradiction.
Thus p/q does not exist.
We now discuss intrinsic approximation in Case 0:
Lemma 3.4. Suppose that Jand x=π(ω)satisfy Case 0. Then for every ψ:N→(0,∞),
a point y=π(τ)−x∈Jxis badly intrinsically approximable with respect to ψif and only
if there exists K < ∞such that for every pair (n, r)∈N2such that ωn+r
n+1 =τn+r
n+1 , we have
r≤K+ Ψ(n),
where
(3.2) Ψ(n) := −logd(ψ(dn)).
Proof. Suppose that yis intrinsically well approximable with respect to ψ. Then for all
ε > 0 there exist infinitely many rational approximations p/q =π(η)−x∈Q∩Jxsatisfying
|y−p/q| ≤ εψ(q)/q. By condition (iii), p/q can be written in the form p/dnfor some
p, n ∈N. On the other hand, since dis prime, it follows that the reduced form of the
fraction p/dnis also of the form p/dn(possibly with different p, n). In other words, qred =dn
for some n∈N. Now since π(η)−π(ω) = p/q, we have that ηagrees with ωexcept for
the first ndigits. On the other hand, we have |π(τ)−π(η)| ≤ εψ(dn)/dn, which implies
that τand ηagree on the first ⌊log1/d (εψ(dn)) + n⌋ − 1 digits (here we are using condition
(ii)). Thus ωn+r
n+1 =τn+r
n+1 , where r=⌊log1/d(εψ(dn))⌋ − 1. Thus for all K < ∞, there exist
infinitely many pairs (n, r)∈N2such that ωn+r
n+1 =τn+r
n+1 but r≥K+ log1/d(ψ(dn)).
On the other hand, suppose that for all K < ∞, there exist infinitely many such pairs.
For each pair (n, r), if we define ηto be the string which agrees with τfor the first ndigits
but then agrees with ω, we find that the rational approximation p/q := π(η)−x∈Q∩Jx
satisfies |y−p/q| ≤ εψ(q)/q.
As an immediate consequence, we get the optimality of the Dirichlet function ψ(q) = 1:
Theorem 3.5 (Optimality).Let Jand xsatisfy Case 0 and let δbe the Hausdorff dimen-
sion of J. Then the set of numbers y∈Jxwhich are badly intrinsically approximable with
respect to the approximation function ψ(q) = Kis of Hausdorff dimension δ.
Proof. We have Ψ(n) = 0, so y=π(τ)−x∈Jxis badly approximable with respect to (3.1)
if and only if the length of the strings on which ωand τagree is uniformly bounded. Thus
for every k∈N, the set
Sk:= {τ∈EN:τkn 6=ωkn ∀n∈N}
is contained in the set of badly approximable points. On the other hand, it is readily
computed that the Hausdorff dimension of Sktends to δas ktends to infinity.
12 L. FISHMAN AND D. SIMMONS
Finally, using a theorem of Levesley, Salp, and Velani [8], we are able to prove the
following theorem, which incorporates both a Khinchin-type and a Jarnik-Besicovitch-type
theorem:
Theorem 3.6. Let Jand xsatisfy Case 0 and fix any function ψ:N→(0,∞). Let
fbe a dimension function such that t7→ t−δf(t)is monotonic. If we denote the set of
ψ-intrinsically well approximable points by WAψ,int, then
HDf(WAψ,int) = (0if P∞
n=1 f(ψ(dn)/dn)(dn)δ<∞
HDf(J)if P∞
n=1 f(ψ(dn)/dn)(dn)δ=∞.
Corollary 3.7. If ψ(q) = ln(q)−1/δ , then almost every point is intrinsically well approx-
imable with respect to ψ; if ψ(q) = ln(q)−(1+ε)/δ, then almost every point is badly intrinsi-
cally approximable with respect to ψ.
Proof of Theorem 3.6. We will use the following theorem from [8]:
Theorem 3.8 ([8] Theorem 1).For any approximation function ψ, consider the set
WAψ,term := {x∈[0,1] : |x−p/q|< ψ(q)for infinitely many (p, q)∈N×dN},
i.e. the set of all points which are ψ-approximable with respect to the rationals with termi-
nating base dexpansions.
Now suppose that Jis a ×d-invariant limit set satisfying (i)-(ii). Let fbe a dimension
function such that t7→ t−δf(t)is monotonic. Then
HDf(WAψ,term ∩J) = (0if P∞
n=1 f(ψ(dn)) ×(dn)δ<∞
HDf(J)if P∞
n=1 f(ψ(dn)) ×(dn)δ=∞.
In [8] the theorem is stated in the case d= 3, Jthe ternary Cantor set, but the proof
clearly generalizes.
Let us call a point y∈Jbadly terminally approximable with respect to ψif y /∈WAεψ,term
for some ε > 0. Clearly, the proof of Lemma 3.4 generalizes to the following statement:
Lemma 3.9. Suppose that Jis as above. Then for every ψ:N→(0,∞), a point y=
π(τ)∈Jis badly terminally approximable with respect to ψif and only if there exists
K < ∞such that for every pair (n, r)∈N2such that τn+r
n+1 is all 0s or all 1s, we have
r≤K+ Ψ(n).
Now let x=π(ω)∈Jbe a Case 0 point, i.e. xsatisfies (iii). Define an automorphism
Φ : EN→ENby the following procedure:
•For each n∈N, choose a permutation Φnof Esuch that Φn(ωn) = 0.
•Let Φ(τ) = (Φn(τn))n.
Clearly, Φ is an isometry of EN, and so the map e
φ:Jx→Jdefined by
e
Φ(y) = π(Φ(x+y))
is bi-Lipschitz. Furthermore, e
Φ sends rational points of Jxto left endpoints of J, whose
denominator is the same up to a constant.
INTRINSIC APPROXIMATION FOR FRACTALS 13
Observe now that a point y∈Jxis badly intrinsically approximable with respect to an
approximation function ψif and only if both e
Φ(y) is badly terminally approximable with
respect to q7→ qψ(q) (the factor comes from a difference in notation between our paper
and [8]). Thus, the Hausdorff measure of the badly intrinsically approximable points agrees
up to a constant with the Hausdorff measure of the badly terminally approximable points.
Applying Theorem 3.8 completes the proof.
4. The intrinsic denominator
In this section we assume that Jis a limit set of a rational IFS satisfying the open set
condition.
Suppose that p/q ∈Q∩Jis the image of the eventually periodic word ω∈EN. Fix
n∈Nso that σn(ω) is periodic, and let mbe a period of σn(ω), so that
ω=ω1. . . ωnωn+1 ...ωn+mωn+1 . . . .
Based on ω,n, and m, we can define p(i),q(i),r(i),i= 1,2 as in the proof of Theorem 2.1
and we define the intrinsic denominator of p/q ∈Jwith respect to the triple (ω, n, m) to
be the denominator of (2.3); i.e. the intrinsic denominator is the number
q(1)(q(2) −p(2) ).
Observation 4.1. If ωis fixed, then the intrinsic denominator is minimized when nand
mare minimal. Furthermore, this intrinsic denominator divides the intrinsic denominator
of p/q with respect to any other pair (en, em).
Thus, we define the intrinsic denominator of p/q with respect to ωto be the intrinsic
denominator with respect to (ω, n, m), with nand mminimal. The intrinsic denominator of
p/q with respect to ωrepresents “everything that the symbolic representation p/q =π(ω)
can tell us about the denominator of p/q”.
In general, a fixed rational number could have more than one eventually periodic symbolic
representation, or it could have none at all. However, we do not know of any examples of
rationals with symbolic representations which are not eventually periodic. Furthermore, in
most of the cases that we care about, this is impossible:
Lemma 4.2. Suppose that pa=±1for all a∈E, where paare given by (1.1). Then every
rational in Jis the image of an eventually periodic word (and therefore has an intrinsic
denominator).
Remark 4.3. In the case where Jis ×d-invariant, then this lemma is well-known (it asserts
that every rational has an eventually periodic d-ary representation). However, the lemma
does not appear to be well-known e.g. for the IFS “1/3,1/4”
u0(x) = x/3
u1(x) = x/4 + 3/4.
Proof of Lemma 4.2. For each a∈E, since pa=±1 we can write
u−1
a(x) = ±(qax−ra).
In particular, if xis rational then the denominator of u−1
a(x) divides the denominator of x.
Thus if ω∈ENis a preimage of xunder π, then the forward orbit (σn(ω))nlies in the set
14 L. FISHMAN AND D. SIMMONS
π−1{p/q :qdivides the denominator of x}, which is finite by the open set condition. Thus
ωis eventually periodic.
For the remainder of this section, we will restrict ourselves to the case where pa=±1
for all a∈E, so that every rational in Jhas at least one intrinsic denominator. We will
also impose the following condition which guarantees that no rational can have more than
one intrinsic denominator:
Definition 4.4. The IFS (ua)asatisfies the strong separation condition if there exists a
closed interval [c, d] such that the collection (ua([c, d]))ais a disjoint collection of subsets
of [c, d].
For example, the IFS for the ternary Cantor set satisfies the strong separation condition.
Note that the strong separation condition implies the open set condition, since we can take
our open set to be (c, d).
From now on, we will assume that our IFS satisfies the strong separation condition. Since
this condition implies that every element of Jhas exactly one symbolic representation, it
follows that every rational in Jhas exactly one intrinsic denominator.
Notation 4.5. The intrinsic denominator of p/q ∈Jwill be denoted qint, whereas the
denominator of p/q in reduced form will be denoted qred.
Observation 4.6. We have qred ↿qint.
Definition 4.7. Let ψ: (0,∞)→(0,∞) be a nonincreasing function. A point x∈Jis
said to be badly symbolically approximable with respect to ψif there exists ε > 0 such that
for all p/q ∈Q∩Jwe have
|x−p/q| ≥ εψ(qint)
qint
.
Otherwise, xis said to be symbolically well approximable with respect to ψ.
So, badly intrinsically approximable implies badly symbolically approximable, but not
vice-versa.
Notation 4.8. If ω∈Eris a finite word, we define the pseudolength of ωto be the number
ℓp(ω) :=
r
X
i=1
ln(qωi).
In the case where the set Jis ×d-invariant for some d, the pseudolength of ωis just equal
to ln(d) times the length of ω.
Lemma 4.9. Suppose that
i) ψis slowly varying i.e. ψ(Kq)≍ψ(q)for all K > 0
ii) ψis bounded
Then for all x=π(ω)∈J,xis badly symbolically approximable with respect to ψif and
only if there exists K < ∞such that for every finite word ηof length rwhich occurs twice
(possibly overlapping) in the initial segment ωℓ
1, we have
(4.1) ℓp(η)≤K+ Ψ(ℓp(ωℓ−r
1)),
INTRINSIC APPROXIMATION FOR FRACTALS 15
where
Ψ(t) := −ln(ψ(et)).
Note that if ψ(q) = ln(q)−s/δ then Ψ(t) = sln(t)/δ.
Proof. Suppose that x∈Jis symbolically well approximable with respect to ψ. Then
for all ε > 0 there exist infinitely many rational approximations p/q ∈Q∩Jsatisfying
|x−p/q| ≤ εψ(qint)/qint. Fix such a p/q, and let τ∈ENbe its preimage. Let n, m ∈Nbe
minimal such that
τ=τ1. . . τn+mτn+1 ...
According to (2.3), the intrinsic denominator of p/q is equal to
qint := n
Y
i=1
qτi! m
Y
i=1
qτn+i±1!≍
n+m
Y
i=1
qτi.
On the other hand, if ℓis the largest integer for which ωℓ=τℓ, then
|x−p/q| ≍
ℓ
Y
i=1
1
qτi
.
Here we have used the strong separation condition to get the lower bound. Thus we have
ℓ
Y
i=1
1
qτi
.ε n+m
Y
i=1
1
qτi!ψ n+m
Y
i=1
qτi!.
Here we have used the slowly varying condition (i).
Since ψis bounded, by choosing εsmall enough we can force Qℓ
i=1(1/qτi)<Qn+m
i=1 (1/qτi),
which implies ℓ > n +m. Thus we have
(4.2) τ=ω1. . . ωn+mωn+1 ...ωn+mωn+1 ...
and in particular
ℓ
Y
i=n+m+1
1
qωi
.εψ n+m
Y
i=1
qωi!.
and taking negative logarithms yields
ℓp(ωℓ
n+m+1)&+−ln(ε) + Ψ(ℓp(ωn+m
1)).
Since ωℓ−m
n+1 =τℓ−m
n+1 =τℓ
n+m+1 =ωℓ
n+m+1, it follows that there are infinitely many words η
which are repeated in ωbut do not satisfy (4.1).
On the other hand, suppose that for all K < ∞there exist infinitely many words ηwhich
are repeated in ωbut do not satisfy (4.1). For each such η, let rbe the length of the η,
and let 0 ≤n < n +mbe the places where it occurs, so that η=ωn+r
n+1 =ωn+m+r
n+m+1. Let τbe
defined by (4.2), and let p/q =π(τ). Then ωand τagree up to at least ℓ:= n+m+rplaces,
and a reverse calculation yields that |x−p/q|.e−Kψ(qint)/qint. Thus xis symbolically
well approximable with respect to ψ.
16 L. FISHMAN AND D. SIMMONS
We now discuss the approximability of a µ-random number x∈J. In the following
discussion xwill always denote a µ-random number, and ωwill denote its preimage under
π. We write Pfor probability and Efor expected value, so that P(x∈S) = µ(S) and
E[f(x)] = Rf(t)dµ(t). In particular, the sequence (ωi)iis a sequence of independent and
identically distributed random variables, whose distribution is given by
P(ωn=a) = q−δ
a=e−δℓp(a).
Lemma 4.10. Fix n, m ∈Nand ℓ0>0. Let
En,m,ℓ0:= {η∈E∗:there exists rsuch that ηn+r
n+1 =ηn+m+r
n+m+1 and such that ℓp(ηn+r
n+1)≥ℓ0}.
Then P(ω∈En,m,ℓ0)≤e−δℓ0.
Remark 4.11. It is possible that r > m, so that ωn+r
n+1 and ωn+m+r
n+m+1 overlap. Thus a naive
independence argument does not work.
Proof of Lemma 4.10. Without loss of generality suppose n= 0. For each η∈Em+Nlet
φ(η) :=
eδℓ0η∈E0,m,ℓ0
0ηN
16=ηm+N
m+1 and η /∈E0,m,ℓ0
eδℓp(ηm+N
m+1 )otherwise
.
Then for any η∈Em, we have φ(η) = 1, and for any η∈Em+N, we have
E[φ(ωm+N+1
1)↿ωm+N
1=η]≤φ(η).
A simple induction therefore yields E[φ(ωm+N
1)] ≤1. The result therefore follows from
Markov’s inequality.
Theorem 4.12 (Khinchin for fractals).Suppose that (ua)aand ψare such that the hy-
potheses (i) - (ii) of Lemma 4.9 are satisfied. Also suppose that ψis nonincreasing.
i) If the series
(4.3)
∞
X
q=1
ln(q)ψ(q)δ
q
converges, then for µ-almost every x∈J,xis badly symbolically approximable with
respect to ψ.
ii) If the series
(4.4)
∞
X
q=1
ln(q)ψ(q)δ
qln(ψ(q))
diverges, then for µ-almost every x∈J,xis symbolically well approximable with respect
to ψ.
In particular, if ψ(q) = ln(q)−(2/δ+ε), then case (i) holds, and if ψ(q) = ln(q)−2/δ , then
case (ii) holds.
INTRINSIC APPROXIMATION FOR FRACTALS 17
Corollary 4.13. Let Cbe the ternary Cantor set and µthe Hausdorff measure in the
Cantor’s set dimension restricted to C. Then for µ-almost every x∈J,xis badly sym-
bolically approximable with respect to ψ(q) = ln(q)−(2/δ+ε)and µ-almost every x∈J,xis
symbolically well approximable with respect to ψ(q) = ln(q)−2/δ .
Proof of Theorem 4.12.
i) Fix Kto be determined. For each n, m ∈Nlet ℓn,m =K+ Ψ(n+m). By Lemma 4.10
we have
(4.5) P [
n,m∈N
En,m,ℓn,m !≤X
n,m∈N
e−δ(K+Ψ(n+m)) =e−δK X
n≥2
(n−1)e−δΨ(n).
If (4.3) converges, then the series
∞
X
n=1
ne−δΨ(n)
also converges. Thus for all ε > 0 there exists K < ∞such that the right hand side
of (4.5) is at most ε. In particular, the probability that ω∈Sn,m∈NEn,m,ℓn,m can be
made arbitrarily small. By Lemma 4.9, this implies that if xis µ-random, then xis
badly intrinsically approximable with respect to ψ.
ii) Let αand βbe the maximum and minimum pseudolengths of a single letter, respec-
tively.
Fix K < ∞. Choose a random ω∈EN. Fix t∈N. For each N∈N, we denote by
s(N) the smallest integer such that
ℓp(ωs(N)−1
N)≥ℓt:= K+ Ψ(α22t+2).
We note that for each N∈N, the string ωs(N)−1
Nlies in the set
Eℓt:= {η∈Er:ℓp(η)≥ℓtbut ℓp(ηr−1
1)< ℓt}.
Consider the event
Et: For all N1, N2distinct with 22t≤Ni< s(Ni)≤22t+2 we have ωs(N1)−1
N16=
ωs(N2)−1
N2.
We note that if (4.1) holds, then Etmust hold for all t∈N, due to our choice of ℓt.
Furthermore, the event Etdepends only on the string ω22t+2 −1
22t, and therefore the events
(Et)tare independent. In what follows, we will prove an upper bound on P(Et).
We begin by dividing ω22t+1 −1
22tinto a sequence of subwords (ωNt,i+1−1
Nt,i )iin the fol-
lowing manner: Let Nt,0= 22t, and if Nt,i has been chosen, then let Nt,i+1 =s(Nt,i).
The sequence (ωNt,i+1−1
Nt,i )iis independent and identically distributed with distribution
P(ωNt,i+1−1
Nt,i =η) = e−δℓp(η).
Now for all η∈Eℓt, we have ℓp(η)≤ℓt+α. Thus ℓp(ωNt,i−1
22t)≤i(ℓt+α) for all i.
Let
Nt=22tβ
ℓt+α.
18 L. FISHMAN AND D. SIMMONS
Then ℓp(ωNt,Nt−1
22t)≤22tβ, and so Nt,Nt−22t≤22ti.e. Nt,Nt≤22t+1. It follows that
the sequence (ωNt,i+1−1
Nt,i )Nt−1
i=0 depends only on the string ω22t+1 −1
22t.
Fix a string τof length 22t+1 . We will prove an upper bound on Etconditioned on
the event ω22t+2 −1
22t+1 =τ, which will then yield the unconditional bound we desire.
If τcontains two identical substrings which are members of Eℓt, then the event
ω22t+2−1
22t+1 =τcontradicts Et, so that P(Et↿ω22t+2−1
22t+1 =τ) = 0.
Otherwise, for each i= 0,...,Nt−1, the probability of the event
Et,i:ωNt,i+1
Nt,i is not equal to any substring of τ
is given by
P(Et,i ↿ω22t+2−1
22t+1 =τ) = 1 −X
η∈Eℓt
substring of τ
e−δℓp(η)
and is therefore bounded above by
1−(22t+1 −ℓt−α)e−δ(ℓt+α).
By independence, it follows that the probability that Et,i holds for all i= 0,...,Nt−1
is bounded above by
(4.6) 1−(22t+1 −ℓt−α)e−δ(ℓt+α)Nt.
On the other hand, if Etholds, it is evident that Et,i holds for all i= 0,...,Nt−1.
Thus the probability of Etgiven ω22t+2−1
22t+1 =τis bounded above by (4.6). Since this
conclusion holds for all τ∈E22t, it follows that the unconditional probability of Etis
bounded above by (4.6).
As noted above, if (4.1) holds for every repeat η, then Etholds for all t. Since the
sequence (Et)tis independent, we have
P \
t∈N
Et!≤Y
t∈N1−(22t+1 −ℓt−α)e−δ(ℓt+α)Nt
≤Y
t∈N
exp −Nt(22t+1 −ℓt−α)e−δ(ℓt+α)
= exp −X
t∈N
Nt(22t+1 −ℓt−α)e−δ(ℓt+α)!.
In particular, if the sum
(4.7) X
t∈N
Nt(22t+1 −ℓt−α)e−δ(ℓt+α)≍X
t∈N
24t
ℓt
e−δℓt
diverges, then the probability that (4.1) holds for every repeat ηis zero. Since the
divergence of the sum will be shown to be independent of K, it follows that if the sum
diverges, then µ-almost every point xis intrinsically well approximable with respect
to ψ.
INTRINSIC APPROXIMATION FOR FRACTALS 19
Write α≤22r−2for some r∈N. Then
X
t∈N
24t
ℓt
e−δℓt≥X
t∈N
24t
K+ Ψ(22t+2r)e−δ(K+Ψ(22t+2r))
≥1
24r+2 X
t≥r
22t22t+2
K+ Ψ(22t)e−δ(K+Ψ(22t))
≥1
3
1
24r+2 X
t≥r
22t+2−1
X
n=22t
n
K+ Ψ(n)e−δ(K+Ψ(n))
≍
∞
X
n=0
⌊en+1⌋ − ⌊en⌋
en
n+ 1
Ψ(n)e−δΨ(n)
≥
∞
X
n=0
⌊en+1⌋−1
X
q=⌊en⌋
ln(q)
qΨ(ln(q))e−δΨ(ln(q))
=
∞
X
q=1
ln(q)
qΨ(ln(q))e−δΨ(ln(q))
=
∞
X
q=1
ln(q)ψ(q)δ
qln(ψ(q))
so if (4.4) diverges then (4.7) diverges as well.
5. Optimality of the bound
In this section, we will restrict ourselves to the case where Jis the ternary Cantor set.
We begin by recalling the following conjecture and proposition from [1]:
Conjecture 5.1 ([1] Conjecture 3.3).If
Sn:= {p/q ∈J: gcd(p, q) = 1,3n−1≤q < 3n}
then for all ε1>0we have
#(Sn)∈O(2n(1+ε1)).
Proposition ([1] Corollary 3.4).Conjecture 5.1 implies that µ(VWAJ) = 0, where
VWAJ:= {x∈J:∃ε > 0∃∞p/q ∈J|x−p/q| ≤ q−(1+ε)}.
As mentioned in the Introduction, we cannot prove Conjecture 5.1 at this time, but we
will reduce it to a simpler conjecture which a heuristic argument suggests is true.
Definition 5.2. Suppose that p/q is a rational number. The period of p/q is the period of
the ternary expansion of p/q, and will be denoted P(p/q).
20 L. FISHMAN AND D. SIMMONS
Theorem 5.3. For every K < ∞, if
S(K)
n:= {p/q ∈J: gcd(p, q) = 1,3n−1≤q < 3n,and P(p/q)≤Kln(q)}
then for all ε1>0we have
#(S(K)
n)∈O(2n(1+ε1)).
We postpone the proof of Theorem 5.3 to the end of this section and proceed to state
the following immediate corollary:
Corollary 5.4. The following conjecture implies Conjecture 5.1, and thus that µ(VWAJ) =
0:
Conjecture 5.5. There exists K < ∞such that
#(Sn\S(K)
n)∈O(2n(1+ε1)).
We will offer a heuristic argument in support of Conjecture 5.5. This argument will in
fact support the following much stronger conjecture:
Conjecture 5.6. For all K > 2/ln(3/2), we have S(K)
n=Snfor all nsufficiently large.
In particular
#(Sn\S(K)
n)∈o(1).
Heuristic argument for Conjecture 5.6. It is easily verified that Conjecture 5.6 is equivalent
to the inequality
(5.1) lim sup
p,q
p/q∈J
q→∞
P(p/q)
ln(q)≤2
ln(3/2).
Our method is to estimate reality using a probabilistic model, and then show that (5.1)
holds with probability one.
We will not specify our model exactly, but we will assume that it has the following
property:
For each p/q ∈Q, the digits of p/q are independent and identically dis-
tributed until they start repeating.
We do not assume any independence of the digits of p/q from the digits of any other
rational, nor any estimate of the distribution of the periods.
Based on this assumption, if p/q ∈[0,1] is fixed then the probability that p/q ∈Jgiven
that P(p/q) = mis (2/3)m. It follows from standard probability theory that
P(p/q ∈Jand P(p/q)≥m)≤(2/3)m.
Fix ε > 0. We have
P(p/q ∈Jand P(p/q)≥(2 + ε) log3/2(q)) ≤q−(2+ε).
INTRINSIC APPROXIMATION FOR FRACTALS 21
For each Q, the probability that there exist p, q with
q≥Q
p/q ∈J
P(p/q)≥(2 + ε) log3/2(q)
is at most X
p/q∈[0,1]
q≥Q
q−(2+ε)=X
q≥Q
q−(1+ε)−→
Q0.
Thus with probability one, there exists Qsuch that for all p, q with q≥Qand p/q ∈J,
we have P(p/q)≤log3/2(q)(2 + ε). Rearranging yields (5.1).
Remark 5.7. The weakest part of this heuristic argument is the fact that the randomness
is not open to a statistical interpretation. We are not saying “If you pick a rational at
random, this should happen” but rather “If you pick a random mathematical universe,
then this should happen” (which of course makes no sense as a logical statement). In fact,
the former statement would be insufficient to support Conjecture 5.6 (or even Conjecture
5.5), since we need that the size of the set of exceptions in proportion to the set of all
rationals in a given range tends to zero exponentially fast.
Proof of Theorem 5.3. Let C4=Kln(3). We have
S(K)
n⊆
C4n
[
m=1
{p/q ∈J: gcd(p, q) = 1, q < 3n,and P(p/q) = m}.
For each q < 3n, we have
#{p= 0,...,q :p/q ∈J} ≤ C52n
by the fractal pigeonhole principle. Thus
#(S(K)
n)≤C52n
C4n
X
m=1
#{q < 3n:∃pgcd(p, q) = 1, P (p/q) = m}.
Fix q∈N, and suppose that there exists pwith gcd(p, q) = 1 and P(p/q) = m. Write
q= 3reqwhere 3 does not divide eq. Then gcd(p, eq) = 1 and P(p, eq) = m. Furthermore
the ternary expansion of p/eqis (immediately) periodic. A simple calculation shows that
p/eq=i/(3m−1) for some i= 0,...,3m−1. Since p/eqis in reduced form, this implies that
eqdivides 3m−1. To summarize:
#{q < 3n:∃pgcd(p, q) = 1, P (p/q) = m} ≤ #{(r, eq) : 0 ≤r < n, eqdivides 3m−1}
=nτ(3m−1),
where τis the number-of-divisors function.
The following result concerning the number-of-divisors function was proven by Ramanu-
jan [10]:
lim sup
N→∞
ln(τ(N))
ln(N)/ln ln(N)= ln(2).
22 L. FISHMAN AND D. SIMMONS
Thus for every ε > 0, we have
τ(N)≤N(ln(2)+ε)/ln ln(N)
for all Nsufficiently large. In particular, if we fix ε2>0 to be determined, then
τ(N)≤Nε2
for all Nsufficiently large. Let C6,ε2be large enough so that
τ(N)≤C6,ε2Nε2
for all N∈N.
Combining our several equations yields
#(S(K)
n)≤C5C6,ε2n2n
C4n
X
m=1
(3m−1)ε2≍n2n3nC4ε2.2n(1+ε1)
if ε2is chosen small enough so that 3C4ε2<2ε1.
References
[1] R. Broderick, L. Fishman, and A. Reich, Intrinsic approximation on Cantor-like sets, a problem
of Mahler, Moscow Journal of Combinatorics and Number Theory, 1, (2011), Issue 4, 291-300.
[2] Y. Bugeaud Diophantine approximation and Cantor sets, Mathematische Annalen, 341, (2008),
no 3, 677-684.
[3] L. Fishman, Schmidt’s game on fractals, Israel J. Math. 171 (2009), no. 1, 77-92.
[4] J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.
[5] D. Kleinbock, E. Lindenstrauss and B. Weiss, On fractal measures and Diophantine approximation,
Selecta Math. 10 (2004), 479-523.
[6] D. Kleinbock and B. Weiss, Badly approximable vectors on fractals, Israel Journal of Mathematics,
149 (2005), 137-170.
[7] S. Kristensen, R. Thorn, and S.L. Velani, Diophantine approximation and badly approximable
sets, Advances in Math. 203 (2006), 132–169.
[8] J. Levesley, C. Salp, and S. Velani, On a problem of K. Mahler: Diophantine approximation and
Cantor sets, Math. Ann. 338 (2007), 97-118.
[9] K. Mahler, Some suggestions for further research, Bulletin of the Australian Mathematical Society,
29 (1984), 101-108.
[10] S. Ramanujan, On the number of divisors of a number, Journal of the Indian Mathematical Society
7 (1915), 131-133]. Collected papers of Srinivasa Ramanujan, 44-46, AMS Chelsea Publications,
Providence, RI, 2000.
[11] W.M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer-
Verlag, Berlin, 1980.
[12] B. Weiss, Almost no points on a Cantor set are very well approximable, Proc. R. Soc. Lond. A
457 (2001), 949–952.
University of North Texas, Department of Mathematics, 1155 Union Circle #311430,
Denton, TX 76203-5017, USA
E-mail address:DavidSimmons@my.unt.edu
E-mail address:lfishman@unt.edu
