Automorphisms with exotic orbit growth

0.29 MB·English

by  Stephan Baier

AUTOMORPHISMS WITH EXOTIC ORBIT GROWTH
STEPHANBAIER,SAWIAN JAIDEE,SHAUNSTEVENS,ANDTHOMASWARD
2
1
0 Abstract. We exhibit continua on two different growth scales in the dynamical
2 Mertens’theorem forergodicautomorphismsofone-dimensionalsolenoids.

n
a
J

1
1. Introduction
2
Automorphisms of compact metric groups provide a simple family of dynamical sys-
]
S tems with additional structure, rendering them particularly amenable to detailed anal-
D ysis. On the other hand, they are rigid in the sense that they cannot be smoothly
perturbed, and for a fixed compact metric group the group of automorphisms is itself
.
h
countable and discrete. Thus it is not clear which, if any, of their dynamical properties
t
a can vary continuously. The most striking manifestation of this is that it is not known
m if the set of possible topological entropies is countable or is the set 0, (this ques-
r 8s
[ tion is equivalent to Lehmer’s problem in algebraic number theory; see Lind [13] or the

monograph[7] for the details). The possible exponential growthrates for the number of
1
closed orbits is easier to decide, and it is shown in [23] that for any C 0, there is a
v
Pr 8s
3 compact group automorphism T :X X with
Ñ
0
5 n1 logFTpnqÝÑC (1)
4
as n , where F n x X :Tnx x . Unfortunately, the systems constructed
1. to acÑhie8ve this contTinpuqum“ |otf dPifferent gr“owtuh|rates are non-ergodic automorphisms of
0
totallydisconnectedgroups,andsocannotbeviewedasnaturalexamplesfromthepoint
2
ofviewofdynamicalsystems. Itisnotclearifaresultlike(1)ispossiblewithinthemore
1
: natural class of ergodic automorphisms on connected groups, unless C is a logarithmic
v
Mahler measure (in which case there is a toral automorphism that achieves this).
i
X Our purpose here is to indicate some of the diversity that is nonetheless possible for
r ergodicautomorphismsofconnectedgroups,forameasureofthe growthinclosedorbits
a
that involves more averaging than does (1). To describe this, let T : X X be a
Ñ
continuous mapon a compact metric space with topologicalentropyh h T . A closed
“ p q
orbit τ of length τ n is a set x,T x ,T2 x ,...,Tn x x with cardinality n.
| | “ t p q p q p q “ u
Following the analogy between closed orbits and prime numbers advanced by work of
Parry and Pollicott [18] and Sharp [21], asymptotics for the expression
1
M N
Tp q“ eh T τ
τ N p q| |
| ÿ|ď
Date:January24,2012.
2010 Mathematics Subject Classification. 37C35,11J72.
SS and TW particularly thank Tim Browning for his endless patience in answering our Analytic
NumberTheoryquestions,andforputtingusintouchwithSB.
SSsupportedbyEPSRCgrantEP/H00534X/1.
1
2 STEPHANBAIER,SAWIANJAIDEE,SHAUNSTEVENS,ANDTHOMASWARD
may be viewed as dynamical analogues of Mertens’ theorem. The expression M N
T
p q
measures in a smoothed way the extent to which the topological entropy reflects the
exponential growth in closed orbits or periodic points. A simple illustration of how M
T
reflects this is to note that if
FT n C1ehn O eh1n
p q“ `
´ ¯
for some h h, then
1
ă
N
1
MT N C1 C2 O e´h2N
p q“ n ` `
nÿ“1 ´ ¯
for some h 0 (see [17]).
2
Writing Oą n for the number of closed orbits of length n, we have
T
p q
F n dO d
T T
p q“ p q
dn
ÿ|
and hence
1
O n µ n F d (2)
Tp q“ n d Tp q
dn
ÿ| ` ˘
by Mo¨bius inversion.
For continuous maps on compact metric spaces, it is clear that all possible sequences
arise for the count of orbits (see [19]; a more subtle observation is that the same holds
in the setting of C diffeomorphisms of the torus by a result of Windsor [25]). For
8
algebraicdynamicalsystemsthesituationisfarmoreconstrained,anditisnotclearhow
muchfreedomthere is inpossible orbit-growthrates. Our purposehere is to exhibit two
different continua of growth rates, on two different speed scales:
for any κ 0,1 there is an automorphism T of a one-dimensional compact
‚ metric grouPppwithq M N κlogN;
T
p q„
for any δ 0,1 and k 0 there is an automorphism T of a one-dimensional
‚ compact mPeptricqgroup wiąth M N k logN δ.
T
p q„ p q
While this plays no part in the argument, it is worth noting that there is a complete
divorcebetweenthetopologicalentropyandthegrowthinclosedorbitsoftheseexamples
– they all have topological entropy log2.
2. The systems studied
We will study a family of endomorphisms (or automorphisms) of one-dimensional
solenoids,allbuilt as isometricextensions of the circle-doublingmap. To describe these,
let P denote the set of rational primes, and associate to any S P the ring
Ă
R r Q: r 1 for all p P S ,
S “t P | |p ď P z u
where denotes the normalized p-adic absolute value on Q, so that p p 1. Thus,
|¨|p | |p “ ´
for example, RH “ Z, Rt2,3u “ Zr61s, and RP “ Q. Let T “ TS denote the endo-
morphism of R dual to the map r 2r on R . This map may be thought of as
S S
ÞÑ
an isometric extension of the circle-doubling map, with topological entropy h T
S
p q “
maxxlog 2 ,0 log2 (see [14] for an explanation of this formula, and for
thpePSsiYmt8pluest extamp|le|spofuho“w the set S influences the number of periodic orbits). Each
ř
element of S destroys some closed orbits, by lifting them to non-closed orbits in the
AUTOMORPHISMS WITH EXOTIC ORBIT GROWTH 3
isometric extension; see [5] for a detailed explanation in the case S 2,3 . This is
“ t u
reflected in the formula for the count of periodic points in the system,
F n 2n 1 2n 1 (3)
TSp q“p ´ q | ´ |p
p S
źP
(see[2]forthegeneralformulabeingusedhere),showingthateachinvertedprimepinS
in the dual groupR removesthe p-partof 2n 1 from the total count of all points of
S
p ´ q
period n. The effect of each inverted prime in R on the count of closed orbits via the
S
relation (2) is more involved.
We write x x for convenience, and since we will be using the same
underlying m|ap|Sth“roughpPoSut|,|wpe will replace T T by the parameter S defining the
S
ś “
system in all of the expressions from Section 1. There are then three natural cases: the
‘finite’ case with S and the ‘co-finite’ case with P S , together producing
| | ă 8 | z | ă 8
countablymanyexamples,andthemorecomplexremaining‘infiniteandco-infinite’case.
A special case of the results in [3] is that for S finite we have
1
MTSpNq“MSpNq“ ehτ “kSlogN `CS `O N´1 ,
τ N | |
| ÿ|ď ` ˘
for some k 0,1 Q and constant C . For example, [3, Ex. 1.5] shows that
S S
Pp sX
269
k .
3,7
t u “ 576
Here we continue the analysis further, showing the following theorem.
Theorem 1. The set of possible values of the constant k with
S
MS N kSlogN CS O N´1 ,
p q“ ` `
as S varies among the finite subsets of P, is dense in `0,1 . ˘
r s
If S is infinite, then more possibilities arise, but with less control of the error terms.
Theorem 2. For any k 0,1 , there is an infinite co-infinite subset S of P with
Pp q
M N klogN.
S
p q„
We also giveexplicit examples ofsets S for which the value ofk arisingin Theorem2
is transcendental.
The co-finite case is very different in that M N converges as N ; other orbit-
S
p q Ñ 8
counting asymptotics better adapted to the polynomially bounded orbit-growth present
in these systems are studied in [4] and [11]. The following result is more surprising, in
that a positive proportion of primes may be omitted from S while still destroying so
many orbits that M N is bounded.
S
p q
Proposition 3. There is a subset S of P with natural density in 0,1 such that
p q
MS N CS O N´1 .
p q“ `
In fact there are such sets S with arbitrarily sm`all n˘on-zero natural density.
WhileitseemshopelesstodescribefullytherangeofpossiblegrowthratesforM N
S
p q
as S varies, we are able to exhibit many examples whose growth lies strictly between
that of the examples in Theorem 2 and that of the examples in Proposition 3.
Theorem 4. For any δ 0,1 and any k 0, there is a subset S of P such that
Pp q ą
M N k logN δ.
S
p q„ p q
4 STEPHANBAIER,SAWIANJAIDEE,SHAUNSTEVENS,ANDTHOMASWARD
We also find a family of examples whose growth lies between that of the examples in
Theorem 4 and of those in Proposition 3.
Theorem 5. For any r N and any k 0, there is a subset S of P with
P ą
M N k loglogN r.
S
p q„ p q
Moreover,itispossibletoachievegrowthasymptotictoanysuitablefunctiongrowing
slower than loglogN. A byproduct of the constructions for Theorems 1 and 5 gives
sets S such that both MS N and MP S N are o logN .
The idea behind the prpoofqs of all thzepse rqesult ips ratheqr similar. We choose our set of
primes S so that it is easy to isolate a subseries of dominant terms in M N in such a
S
p q
waythatthesumoftheremainingtermsconverges,usuallyquickly(controllingthisrate
governs the error terms). We describe a general framework for dealing with such sets,
andthenthe sets usedto carryoutthe constructionsaredefinedby arithmeticalcriteria
relying on properties of the set of primes p for which 2 has a given multiplicative order
modulo p.
Notation. From (3), we have M N M N , for any set S: thus, without loss of
S S 2
generality,we make the standingpassqu“mptioYntthupatq2 S. We will use various globalcon-
R
stants C ,C ,..., each independent of N and numbered consecutively. The symbols C
1 2
and C denote local constants specific to the statement being made at the time. For an
S
odd prime p, denote by m the multiplicative order of2 modulo p; for a set T ofprimes,
p
we write m lcm m :p T . We will also use Landau’s big-O and little-o notation.
T p
“ t P u
3. Asymptotic estimates
By (2) and (3) we have
1
M N µ n 2d 1 2d 1
Sp q“ n2n d ´ ˆ ´ S
n N dn
ÿď ÿ| ` ˘ˇ ˇ ˇ ˇ
2n 1 ˇ ˇ ˇ ˇ
| ´ |S R N ,
S
“ n ` p q
n N
ÿď
“FSpNq
where the last equation defineslobooootohoomFooooNooonand R N .
S S
p q p q
Lemma 6. R N C O 2 N 2 .
S S ´ {
p q“ `
Proof. By definition, R N is`the su˘m of two terms,
S
p q
N 2n 1 N 1
R N | ´ |S µ n 2d 1 2d 1 .
Sp q“´ n2n ` n2n d p ´ q ´ S
n 1 n 1 dn,d n
ÿ“ ÿ“ |ÿă ` ˘ ˇ ˇ
2n 1 1 1 1 ˇ ˇ
Since | ´ |S and 2d 1 2d 1 , both sums converge
n2n ď 2n n2n p ´ q ´ S ď 2n 2
dn,d n {
|ÿă ˇ ˇ
(absolutely) so RS N converges to some CS. Mˇoreoverˇ,
p q
R N C 8 1 8 1 O 2 N 2 .
| Sp q´ S|ď 2n ` 2n 2 “ ´ {
n“ÿN`1 n“ÿN`1 { ´ ¯
(cid:3)
AUTOMORPHISMS WITH EXOTIC ORBIT GROWTH 5
Thus we think of F N as a dominant term, and much of our effort will be aimed at
S
p q
understanding how F N behaves as a function of S, which starts with understanding
S
p q
the arithmetic of 2n 1. The main tool here is the elementary observation that, for p a
´
prime and n N,
P
ord 2n 1 ordpp2mp ´1q`ordppnq if mp |n, (4)
p
p ´ q“#0 otherwise,
where ord n denotes the index of the highest power of p dividing n, so that n
p p
p q | | “
p´ordppnq. In particular, if T is a finite set of primes and n N, then
P
2nmT 1 2mT 1 n . (5)
| ´ |T “| ´ |T | |T
In the proof of [3, Proposition 5.3], a recipe is given for computing the coefficient
of logN in the asymptotic expansion of F N , when S is finite. This is based on an
S
p q
inclusion-exclusionargument,splitting upthe sum F N accordingto the subsets ofS.
S
p q
The disadvantage of this approachis that many subsets of S can lead to an empty sum:
in principle, the splitting works for infinite S (since F N is anyway a finite sum) but
S
p q
then the decomposition of the sum falls into an uncountable number of pieces. Here we
takeadifferentapproach,splittinguptheexpressionsarisingaccordingtothevaluesm ,
T
for T a finite subsetof S,ratherthanaccordingto the subsets T. In this setting, several
different subsets T may give the same value for m .
T
To this end, we set M m : p S and denote by M its closure under taking
S p S
“ t P u
least common multiples:
Ď
M lcm M :M M m :T S .
S 1 1 S T
“t p q Ď u“t Ď u
For n N, we put
P Ď
m¯ max m¯ M :m¯ n lcm m M :m n .
n S p S p
“ t P | u“ t P | u
Then, for m¯ M, set
P Ď
N n N:m¯ m¯
m¯ n
“t P “ u
Ď
and
S p S :m m¯ .
m¯ p
“t P | u
Note that the sets S are finite, even if S is not. Then, using (5), we get
m¯
2n 1
F N | ´ |S
S
p q “ n
m¯PÿMS nnÿPďNNm¯
Ď |2m¯ ´1|Sm¯ |n|Sm¯ . (6)
“ m¯ n
m¯PÿMS mp∤nm¯nďÿfoNr{pm¯PSzSm¯
The asymptotic behaviour of thesĎe inner sums can be computed, at least in principle,
usingtheresultsof[3,§5]. However,forageneralsetofprimesS,thisiscumbersome,so
we will specialize to sets which are easier to deal with. In this, we are motivated by the
next lemma, which follows one of the many paths used to prove Zsigmondy’s theorem
(see [6, § 8.3.1] for the details).
Lemma 7. Fix n N, and let S P be a set of primes containing p P : m n .
p
P Ă t P “ u
Then
n
2n 1 .
| ´ |S ď 2φ n 2
p q´
6 STEPHANBAIER,SAWIANJAIDEE,SHAUNSTEVENS,ANDTHOMASWARD
Recall that a divisor of 2n 1 is primitive (in the sequence 2n 1 ) if it has no
common factor with 2m 1, f´or any m with 1 m n. Thusp p ´Pq:něm1 n is the
p
´ ď ă t P “ u
set of primitive prime divisors of 2n 1. This set is finite, since m log p, but may
´ p ě 2
be large – for example m m m 29. Schinzel [20] proved that there are
233 1103 2089
“ “ “
infinitely many n for which this set contains at least 2 elements, but it seems that not
much more is known about it in general.
Proof of Lemma 7. Writing 2n 1 for the maximal primitive divisor of 2n 1, we
˚
p ´ q ´
certainly have
|2n´1|´S1 ěp2n´1q˚. (7)
By factorizing xn 1 we have
´
2n 1 Φ 2 , (8)
d
´ “ p q
dn
ź|
whereΦ is the dthcyclotomicpolynomial. Itfollowsthat 2n 1 isa factorofΦ 2 .
d ˚ n
p ´ q p q
If a prime p divides gcd Φ 2 ,Φ 2 for some d n with d n, then p 2d 1. Then,
n d
p p q p qq | ă | ´
from (4),
ord 2n 1 ord 2d 1 ord n d
p p p
p ´ q“ p ´ q` p { q
and, from (8),
ord 2n 1 ord 2d 1 ord Φ 2 ord 2d 1 1,
p p p n p
p ´ qě p ´ q` p p qqě p ´ q`
so in particular p divides n d; therefore p divides n and d divides n p, so p divides
{ {
2n{p 1 .
p ´ q
Moreover
ord 2n 1 ord 2n p 1 1
p p {
p ´ q“ p ´ q`
and
ordp 2n 1 ordp 2n{p 1 ordp Φn 2 ,
p ´ qě p ´ q` p p qq
so in fact ord Φ 2 1. Thus gcd Φ 2 , Φ 2 divides p, which is
pp np qq “ np q d|n,dăn dp q p|n
at most n, and ´ ¯
ś ś
2n 1 ˚ Φn 2 n. (9)
p ´ q ě p q{
On the other hand, by Mo¨bius inversion applied to (8),
Φn 2 2d 1 µpn{dq
p q“ p ´ q
dn
ź|
so
log Φn 2 φ n log 2 µ n d log 1 2´d ,
p p qq“ p q p q` p { q p ´ q
dn
ÿ|
where φ n is the Euler totient function. Now, using the Taylor expansion for the loga-
p q
rithm,
µ n d log 1 2 d 8 2´jd 8 2´j 2 j d 1 2log2,
´ ´ p ´ q
ˇ p { q p ´ qˇď j “ j ď
ˇdn ˇ dnj 1 j 1 dn
ˇÿ| ˇ ÿ| ÿ“ ÿ“ ÿ|
ˇ ˇ
so Φ 2 ˇ 2φ n 2 and the resuˇlt follows by (7) and (9). (cid:3)
np qˇě p q´ ˇ
AUTOMORPHISMS WITH EXOTIC ORBIT GROWTH 7
This lemma will be used as follows. Instead of starting with a set S of primes, we
begin with M a subset of N and put
SM p P:mp M .
“t P P u
Then
2n 1 2n 1 2n 1
| ´ |SM | ´ |SM | ´ |SM,
n “ n ` n
n N nďN nďN
ÿď nÿRM nÿPM
“DSMpNq “QSMpNq
and, by Lemma 7, looooooooomooooooooon looooooooomooooooooon
1
Q N C ,
SMp qď 3 2φ n
n N p q
ÿď
whichconvergessinceφ n ?n,forn 6. Moreover,the sameobservationshowsthat
p qě ě
QSMpNq“C4`O 2´?N .
´ ¯
Thus the asymptotic behaviour is governed by the dominant term D N . From
SMp q
Lemma 6, we get
2n 1
MSMpNq“ | ´n |SM `C5`O 2´?N . (10)
nnÿRďMN ´ ¯
All our examples will take this form.
Remarks 8. (i) We have set up maps S MS and M SM between the power sets
ÞÑ ÞÑ
ofP 2 and N, whichareorder-preservingfor inclusion. It is easy to check that SM
zt u S Ě
S, while SMSM “SM. Similarly, we have MSM “Mzt1,6u, since all but the first and
sixth terms of the Mersenne sequence have primitive divisors, and M M . In
particular, we can apply the decomposition (6) of M N in tandem wSMithS (“10). WS hen
SMp q
we do so, we will replace M by the closure M of M under least common multiples
SM
to get
2m¯Ď 1 Ď n
MSMpNq“ | ´m¯ |Sm¯ | n|Sm¯ `C6`O 2´?N . (11)
m¯ÿPM mpn∤nďm¯N{fmo¯ÿr,pnPmS¯MRMzSm¯ ´ ¯
(ii) For a general set SĎ, let Mo be the set of m M for which S contains all primes p
S P S
with mp m, and put So SMo; this is the largest subset of S of the form SM.
“ “ S
Similarly,putS SM ,thesmallestsupersetofS ofthe formSM. Thetechniqueshere
“ S
can be applied to the sets So and S and, since
s M N M N M N ,
Sp sqď Sp qď Sop q
we get some information on the asymptotic behaviour of M N .
s Sp q
The formula (11) is particularly simple in the case that M is closed under multipli-
cation by N: that is, if a M and b N, then ab M. In this case M is closed under
P P P
least common multiples and, for n M, we have m¯ 1. Thus the inner sum in (11) is
n
R “
empty for m¯ 1, and we get
‰
1
MSMpNq“ n `C7`O 2´?N . (12)
nnÿRďMN ´ ¯
8 STEPHANBAIER,SAWIANJAIDEE,SHAUNSTEVENS,ANDTHOMASWARD
Provided M N as N , this implies that
SMp qÑ8 Ñ8
1
M N .
SMp q„ n
nďN
nÿRM
Many, though not all, of our examples will be of this form. The task is then to choose
sets M which are closed under multiplication by N, and for which we can control the
asymptoticsofthesumin(12). Onetechniquewewilloftenuseforthisispartial (orAbel)
summation: if we write πM x n x : n M and f is a positive differentiable
p q “ |t ď R u|
function on the positive reals, then
x
f n πM x f x πM t f1 t dt,
p q“ p q p q` p q p q
nďx ż1
nÿRM
with the dominant term generally coming from the integral. In severalcases the asymp-
totics of πM x are already well understood.
p q
4. Finite sets of primes
In order to prove Theorem 1, we need to choose finite sets of primes S for which we
can make good estimates for the coefficient of the leading term in Mertens’ Theorem.
These calculations are simplified by considering only primes p for which m is prime.
p
Let L be a finite set of primes and take M L, so that
“
S SL p P:mp L ,
“ “t P P u
which is a finite set. By [3, Theorem 1.4], we have
MSLpNq“kLlogpNq`CL`O N´1 ,
for some kL 0,1 Q and constant CL. The followin`g lem˘ma gives upper and lower
P p sX
bounds for kL.
Lemma 9. Let L be a finite subset of P.
1 1
(i) We have kL 1 .
ď ´ ℓ ` ℓ 2ℓ 1
ℓ Lˆ p ´ q˙
(ii) For ℓPPzL, we hźPave 1´ 1ℓ kL ď kLYtℓu.
Proof. For L a subset of L, w`e write˘m L ℓ. We break up the Mertens sum
as in (11), no1ting that M m L :L p L1q “: ℓPL1
1 1
“t p q Ď u ś
2m L1 1
MSLĎpNq„Lÿ1ĎLˇˇˇ pmqpL´1qˇˇˇSL1 ℓ∤nnďfNoÿr{mℓPpLLz1Lq1 |n|nSL1. (13)
By [3, Proposition 5.2], we have
n
| |SL1 k logN C O N 1 ,
n “ L11 ` L11 ` ´
n N
ÿP ` ˘
with k p . Moreover,by [3, Lemma 5.1],
L11 “ pPSL1 p`1
ś n ℓ
| |SL1 k 1 | |SL1 logN C O N 1 .
nďN n “ L11ℓ L L1˜ ´ ℓ ¸ ` L21 ` ´
ℓ∤nfoÿrℓPLzL1 Pźz ` ˘
AUTOMORPHISMS WITH EXOTIC ORBIT GROWTH 9
In particular, the coefficient of the logN term is 1 1 p , which
is at most 1 1 . Moreover, for ℓ L aℓnPdLzLp1suc´h ℓthatpmPSL1zL pℓ`,1we have
ord 2m L1 ℓ1PLzL1ord´2ℓℓ 1 so P 1ś ` ˘ś p “
pp p qś´ qě ` pp ˘´ q
1
2m L1 1 .
ˇ p q´ ˇSL1 ďℓźPL1 2ℓ´1
ˇ ˇ
Putting everything back intoˇ(13), we sˇee that the coefficient kL of the logN term is
bounded above by
1 1 1 1
1 1 .
ℓ 2ℓ 1 ´ ℓ “ ´ ℓ ` ℓ 2ℓ 1
L1 Lℓ L1 p ´ qℓ L L1ˆ ˙ ℓ Lˆ p ´ q˙
ÿĎ źP Pźz źP
This proves (i), and the proof of (ii) is similar but easier: we have
2m L1 1
p q n
M N ´ SL1 | |SL1,
SLYtℓup q„L1ĎÿLYtℓuˇˇˇ mpL1qˇˇˇ gcdpnn,ďℓNmÿ{pmLqpqL|m1qpL1q n
and, for L a subset contained in L, the contribution of the sum corresponding to L
1 1
ℓ
is 1 | |SL1 times the contributionof the sum correspondingto L in (13). In partic-
´ ℓ 1
ˆ ˙
ular, the Mertens sum for L ℓ is at least 1 1 times that for L. (cid:3)
Yt u ´ ℓ
Proof of Theorem 1. Letk 0,1 andε 0,`andch˘oosetwoprimesℓ 1 k andℓ
Pp q ą 0 ą `ε 1 ą
ℓ such that 1 1 1 k; this is possible since the product over all
0 ℓ0ďℓăℓ1 ´ l ` lp2l´1q ă
primes greateśr than ℓ0´converges to 0. ¯
We choose recursively a subset L of ℓ P : ℓ ℓ , using the greedy algorithm
1
t P ă u
as follows. Let ℓ P and suppose we have already defined L ℓ : L 1,...,ℓ 1 .
P p q “ Xt ´ u
If k kL ℓ k ε then we are done and L L ℓ ; otherwise ℓ L if and only
if kLďℓ ℓp q kă. ` “ p q P
ThpeqYctlauimě is then that, for the subset L given by this algorithm, the leading coeffi-
cient kL satisfies k kL k ε. The first inequality is clear from the definition, while
ď ă `
the second follows from the following two observations:
(i) There is a prime ℓ with ℓ ℓ ℓ such that ℓ L: if not, by Lemma 9(i),
0 1
ď ă R
1 1
kL 1 k,
ď ´ ℓ ` ℓ 2ℓ 1 ă
ℓ0ďźℓăℓ1ˆ p ´ q˙
which is absurd.
(ii) With ℓ as in (i), we have kL ℓ ℓ k, since ℓ L; thus, by Lemma 9(ii),
p qYt u ă R
ℓ ℓ
0
kLpℓq ďˆℓ´1˙kLpℓqYtℓu ăˆℓ0´1˙k ăk`ε.
(cid:3)
Remark 10. Let L be an infinite set of primes such that 1 diverges and put S
SL “tpPP:mp PLu. ThenMSpNqďMSL1pNq,foranyfinřitℓePLsuℓbsetL1 ofL,soMSpN“q
grows no more quickly than kL1logN. Since kL1 is at most ℓPL1 1´ 1ℓ ` ℓp2ℓ1´1q , by
Lemma9(i),therearefinitesubsetsL1 ofLwithkL1 arbitrariślyclos´eto0,andwede¯duce
that M N o logN .
S
p q“ p q
10 STEPHANBAIER,SAWIANJAIDEE,SHAUNSTEVENS,ANDTHOMASWARD
5. Logarithmic growth for infinite sets of primes
In this section we will prove Theorem 2. Fix ℓ P, and let M n N : ℓ n , so
ℓ
P “ t P | u
that
S S p P:ℓ m . Ď
M ℓ p
ℓ “ “t P | u
By Hasse [9, 10] these sets have a positive Dirichlet density within the set of primes:
for ℓ 2 the density is ℓ , aĎnd forsℓ 2 the density is 17. They also have natural
ą ℓ2 1 “ 24
density by, for example, [24´, Theorem2]. Noting that M is closedunder multiplication
ℓ
by N, by (12) we have
Ď
1
MSℓpNq“ n `C9`O 2´?N
s nÿℓď∤nN ´ ¯
“ 1´ 1ℓ logN `C8`O N´1 .
Indeed, if L is any finite set of pri`mes, a˘pplying the same`argum˘ent to
ML n N:ℓ n for some ℓ L ,
“t P | P u
and SL “tpPP:mp PMLĎu, we get
s MSLĎpNq“ 1´ 1ℓ logN `C10`O N´1 .
ℓ L
źP ` ˘ ` ˘
s
Since the set 1 1 :L P finite is dense in 0,1 , this gives an easy way of
getting a dense sℓePtLof ´valℓues foĂr the leading coefficienrt ins Mertens’ Theorem. Note
ś ` ˘ (
howeverthatTheorem1wasmoredelicate,sincetheclaimwasthatadensesetofvalues
can be obtained using only finite sets S. Similarly, Theorem 2 claims more: every value
in 0,1 can be obtained as leading coefficient.
p q
Proof of Theorem 2. Now let L P be any set of primes for which the product kL :
toℓoPbLta1in´ 1ℓ is non-zero, defineĂML and SL as above, and apply the argument abov“e
ś ` ˘ 1
ĎM N s .
SLp q„ n
nďN
s nRÿML
Applying [22, Theorem I.3.11] we have that nĚ x : n ML kLx thus, by partial
|t ď R u| „
summation, we get
MSLpNq„kLlogN. Ď
This gives Theorem 2 since kL :L P 0,1 . (cid:3)
t Ďs u“r s
6. Sublogarithmic growth
Now we consider sets giving intermediate sublogarithmic growth,provingTheorem4.
We return to sets close to ML and SL of §5 but now for infinite sets of primes L such
that 1 1 0. We will need a result from analytic number theory that allow
sets of pℓPrLimes´toℓbe“selectedĎwith presscribed properties, whose proof we defer to §10.
ś ` ˘
Proposition 11. For any δ 0,1 , there is a set of primes L such that
Pp s
logℓ
δlogx O 1 (14)
ℓ “ ` p q
ℓďx
ÿlPL