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arXiv:1810.13363v1 [math.NT] 31 Oct 2018
ON THE LEHMER CONJECTURE AND COUNTING IN FINITE
FIELDS
EMMANUEL BREUILLARD AND P´
ETER P. VARJ ´
U
Abstract. We give a reformulation of the Lehmer conjecture about algebraic
integers in terms of a simple counting problem modulo p.
Let Sdbe the set of all polynomials of degree at most dwith coefficients in
{0,1}. Recall that if αis an algebraic number, with minimal polynomial πα(X) =
a0Qn
1(X−αi) in Z[X], then the Mahler measure of αis the quantity:
M(α) = |a0|
n
Y
1
max{1,|αi|}.
In [BV16] we have shown that the Mahler measure is related to the growth rate
of the cardinality of the set
Sd(α) := {P(α); P∈ Sd}
as follows:
Theorem 1 ([BV16, Thm 5, Lem 16]).There is a numerical constant c > 0such
that for every algebraic number α,
min{2, M (α)}c≤lim
d→+∞|Sd(α)|1
d≤min{2, M (α)}.
One can take c= 0.44. The limit above always exists by sub-multiplicativity of
d7→ |Sd(α)|. The celebrated Lehmer conjecture posits the existence of a numerical
constant c0>0 such that every algebraic number α, which is not a root of unity
must have
M(α)>1 + c0.
In particular the following reformulation of the Lehmer conjecture follows im-
mediately from Theorem 1.
Corollary 2. The following are equivalent
(1) there exists c1>0such that
lim
d→+∞|Sd(α)|1
d>1 + c1
for every complex number α, which is not a root of unity,
(2) the Lehmer conjecture is true.
Date: November 1, 2018.
Key words and phrases. finite fields, Lehmer conjecture, Mahler measure.
EB acknowledges support from ERC Grant no. 617129 ‘GeTeMo’; PV acknowledges support
from the Royal Society.
1
2 EMMANUEL BREUILLARD AND P ´
ETER P. VARJ´
U
Note that if αis transcendental, or simply not a root of a polynomial of the form
P−Q,P, Q in Sdfor some d, then |Sd(α)|= 2d+1 for all d. Note further that if α
is a root of unity, then |Sd(α)|grows at most polynomially in d.
The set Sd(α) is the support of the random process Pd
0ǫiαiwhere the ǫiare
Bernoulli random variables equal to 0 or 1 with probability 1
2. The proof of the
lower bound in Theorem 1 consists in establishing a lower bound on the entropy of
the above process.
The purpose of this short note is to give a mod pversion of the above equivalence,
showing that Lehmer’s conjecture is equivalent to an easy to formulate assertion
about finite fields.
We first propose the following reformulation. We write log for the base 2 loga-
rithm. Given C≥1, we say that a prime pis C-wild if there is x∈F×
pof mul-
tiplicative order at least (log p)2such that not every element of Fpis a sum of at
most (log p)Celements from the geometric progression H:= {x, x2,...,x[Clog p]},
where [y] denotes the integer part of y. (Here we allow choosing the elements of H
multiple times.)
Theorem 3. The following are equivalent.
(a) The Lehmer conjecture is true.
(b) For some C > 1, almost no prime is C-wild, namely as X→+∞
|{p≤Xand pis C-wild}| =o(|{p≤X}|).
The implication (b)⇒(a) is the easier one. Inasmuch this result can hardly been
seen as making any progress towards a positive answer to the Lehmer conjecture.
The main point of this note is to show the converse implication. A crucial ingredient
in the proof is the lower bound for |Sd(α)|in Theorem 1.
As it turns out the method of proof actually yields a more precise result: the
validity of Lehmer’s conjecture implies a stronger statement than (b), and vice versa
a weaker version of (b) already implies Lehmer’s conjecture. We spell this out in
Theorem 4 below and to this aim we first introduce some notation.
Let δ, ǫ > 0. We will say that a prime number pis δ-bad if there is a non-zero
residue class xmodulo pwith multiplicative order at least log plog log log psuch
that |Sd(x)| ≤ pδfor all d≤log p, where
Sd(x) := {P(x); P∈ Sd}.
For K∈N, we let SK
dbe the family of polynomials of degree at most dwith
integer coefficients in the interval [−K, K]. Clearly
(1) Sd⊂ SK
d⊂ Sd±...± Sd⊂ S2K
d,
where there are 2Ksummands in the sumset.
We will say that pis (δ, ǫ)-very bad if there is a non-zero residue class xmodulo
pwith multiplicative order at least qp
log psuch that |SK
d(x)| ≤ pδfor all d≤1
δlog p
and all K≤2ǫd.
We can now state our second reformulation:
Theorem 4. The following are equivalent.
(a) The Lehmer conjecture is true.
(b) There is δ > 0such that for all ǫ > 0as X→+∞,
(2) |{p≤Xand pis (δ, ǫ)-very bad}| =o(|{p≤X}|).
ON THE LEHMER CONJECTURE AND COUNTING IN FINITE FIELDS 3
(c) For every ε > 0there is δ > 0such that as X→+∞,
(3) |{p≤X;pis δ-bad}| ≪εXε.
We have used the Vinogradov notation ≪ǫto mean that the inequality holds up
to a multiplicative constant depending only on ǫ.
It is clear that (c) implies (b) in the above theorem, because for δ, ǫ ∈(0,1
2) every
large enough (δ, ǫ)-very bad prime must be δ-bad.
The relevance of the Lehmer conjecture to the study of the family of polynomials
Sdis well-known and appears for example in Konyagin’s work [K92, K99].
As we mentioned above, the growth of |Sd(α)|is intimately related to the be-
havior of the random process Pd
0ǫiαiwhere the ǫiare Bernoulli random variables
equal to 0 or 1 with probability 1
2. These processes have been studied by Chung,
Diaconis and Graham [CDG87] and Hildebrand [Hil90]. Very recently they have
been used to study irreducibility of random polynomials in [BV18].
In the remainder of this paper we prove the above theorems. We first handle
Theorem 4 and then deduce Theorem 3.
Acknowledgments. PV wishes to thank Boris Bukh for interesting discussions
on subjects related to this note. In particular, PV first learned about a variant of
the implication (b)⇒(a) in Theorem 3 from him.
1. Proof that (b)implies (a)in Theorem 4
We recall the following fairly well-known lemma (see Lemma 16 in [BV16] and
§8 in [B07]).
Lemma 5. Let αbe an algebraic number. Then there is a constant Cα>0such
that for every d, K ≥1,
|SK
d(α)| ≤ Cα(Kd)CαM(α)d.
If Lehmer’s conjecture fails, then for each δ > 0 there is an algebraic unit αsuch
that M(α)<1 + δ2/2. By the lemma above we conclude that if we choose ǫin the
interval (0, C −1
αδ2/4), then for all large enough dwe have:
|S2ǫd
d(α)| ≤ 2δ2d.
Let Fbe the Galois closure of Q(α). Let pbe a prime number which splits
completely in Fand let pbe a prime ideal in Fabove p. Denoting by ¯αthe residue
class of αmodulo pwe must have:
|S2ǫd
d(¯α)| ≤ 2δ2d,
for all d∈N. In particular if d≤1
δlog p, we get
|S2ǫd
d(¯α)| ≤ pδ.
In particular if the multiplicative order of ¯αis at least qp
log p, then pis (δ, ǫ)-very
bad.
On the other hand, the Frobenius-Chebotarev density theorem tells us that there
is a positive proportion of primes pwhich split completely in F. Therefore in order
to complete the proof, it is enough to show that there is only a vanishing proportion
of primes pfor which ¯αhas multiplicative order at most qp
log p, namely:
4 EMMANUEL BREUILLARD AND P ´
ETER P. VARJ´
U
Lemma 6. Let F|Qbe a finite Galois number field. Let OFbe its ring of integers
and let α∈ OF\ {0}. Let Pαbe the set of primes p∈Nsuch that there is a
prime ideal pin Fabove psuch that |OF/p|=pand αmod pis non-zero and of
multiplicative order in F×
pat most qp
log p. Then as X→+∞,
|{p≤X;p∈ Pα}| =o(|{p≤X}|).
Proof. Note that if αn−1∈p, then p=NF|Q(p) divides NF|Q(αn−1). In particular
Y
p∈Pα;√X≤p≤X
pdivides Y
n≤√X/ log X
NF|Q(αn−1).
On the other hand:
|NF|Q(αn−1)| ≤ Y
σ∈Gal(F|Q)
(1+|σ(α)|n)≤Y
σ∈Gal(F|Q)
2 max{1,|σ(α)|}n≤2[F:Q]M(α)n
We consider this inequality for n= 1,...,(X/ log X)1/2and get
|Pα∩[√X, X]|log X≪(X
log X)1/2[F:Q] + X
log Xlog M(α).
The conclusion follows immediately given that |{p≤X}| ≫ X/ log X.
Remark 7. Clearly we can replace (p/ log p)1/2by p1/2ǫ(p) for any function ǫ(p)
tending to 0 as p→+∞with the same proof. Erd˝os and Murty improve this
even further to pǫ(p) in [EM99] in the special case F=Qassuming the Riemann
hypothesis for certain Dedekind zeta functions.
2. Proof that (a)implies (c)in Theorem 4
The proof of this implication uses the harder part (the lower bound) in Theo-
rem 1.
Let Pdbe the set of polynomials with degree at most dand coefficients in
{−1,0,1}. Note that Pd=Sd− Sd. Let Idbe the set of monic Q-irreducible
polynomials dividing at least one non-zero polynomial in Pd. We will say that
a prime pis d-exceptional if it divides the resultant Res(D1, D2) of some pair of
distinct irreducible polynomials D1, D2in Id.
Recall that given a field Fevery field element α∈Fdetermines a partition πα,d
of Sdmade of the preimages of the map P7→ P(α) from Sdto F. Clearly |Sd(α)|
is the number of parts of πα,d.
Claim 1. If pis not d-exceptional, then for every α∈Fpthere is at most one D∈ Id
such that D(α) = 0 and moreover πα,d =πβ,d if β∈Qis a root of D.
Proof. If pis not d-exceptional, then the reductions mod pof the polynomials in
Idare pairwise relatively prime. In particular they cannot have a common root in
Fp. So each α∈Fpcan be the root of at most one D∈ Id. If there is such a D,
it is unique, and for any P, Q ∈ Sdwe have the equivalences : P(α) = Q(α) if and
only if Ddivides P−Qand if and only if P(β) = Q(β), where βis chosen to be a
complex root of D. This ends the proof.
ON THE LEHMER CONJECTURE AND COUNTING IN FINITE FIELDS 5
Note that if πα,d =πβ,d , then |Sn(α)|=|Sn(β)|for all n≤d. Similarly for any
n≤dwe have αn= 1 in Fpif and only if βn= 1 in Q. Here we used that the n’th
cyclotomic polynomial is in Idfor n≤d.
Claim 2. For dlarge enough, there are at most 10dd-exceptional primes.
Proof. If D1, D2are two polynomials in Id, their resultant can be bounded above
by Hadamard’s inequality:
|Res(D1, D2)| ≤ kD1kdeg D2
2kD2kdeg D1
2,
where kPk2denotes the ℓ2-norm of the coefficients of a polynomial P. Recall
that for every P∈C[X] we have kPk1≤2deg PM(P). This is easily seen by
expressing the coefficients of Pas sums of products of roots of P. Recall also that
M(P)≤ kPk1as can be seen using the Jensen formula for M(P). However if
D∈ Id, then Ddivides some P∈ Pdand hence M(D)≤M(P)≤ kPk1≤d+ 1,
and thus
kDk2≤ kDk1≤2d(d+ 1).
It follows that
|Res(D1, D2)| ≤ 22d2(d+ 1)2d≤24d2.
The number of distinct prime factors of |Res(D1, D2)|is thus at most 4d2when d
is large enough. Since there are at most dirreducible factors of Pfor any P∈ Pd,
|Id| ≤ d3d+1. Hence there can only be at most 4d2|Id|2≤4d49d+1 d-exceptional
primes.
We are now ready to conclude the proof of the implication (c)⇒(a) in Theorem
4. We assume that Lehmer’s conjecture holds inasmuch as there is δ0>0 such that
M(β)> eδ0for every algebraic number β, which is not a root of unity. Let δ > 0
and and X≥1. Assume that δ < (cδ0/2)2, where cis the constant from Theorem
1. Let dbe the integer part of √δlog X.
Claim 3. Let p∈[√X, X ] be a prime number. If pis δ-bad, then pis d-exceptional.
Proof. If pis δ-bad, there is α∈Fp\{0}of multiplicative order at least log plog log log p
such that |Sn(α)| ≤ pδfor all n≤log p. In particular
|S[1
2log X](α)| ≤ Xδ.
On the other hand if pis not d-exceptional, by Claim 1 above, there is β∈Qof
degree at most dsuch that |Sn(α)|=|Sn(β)|for all n≤d. In particular if say
δ < 1
4, we have d≤1
2log Xand thus
|Sd(β)| ≤ Xδ≤e√δ(d+1) ≤e2√δd.
However according to Theorem 1, max{2, M (β)}c≤ |Sd(β)|1/d (this holds for
all dby sub-multiplicativity of d7→ |Sd(β)|). Hence
M(β)≤e2
c√δ.
Since we assume the validity of the Lehmer conjecture, it follows that βmust
be a root of unity. However βis a root of a polynomial Pin Pd, hence has degree
at most dover Q. Its minimal polynomial must be a cyclotomic polynomial Φm
6 EMMANUEL BREUILLARD AND P ´
ETER P. VARJ´
U
dividing Phence in Id. But pis not d-exceptional, so by Claim 1 we know that
Φm(α) = 0.
It is well known that there is an absolute numerical constant C > 0 such that
if xis a root of unity in Qand has degree at most das an algebraic number over
Q, then the order of xis at most Cd log log d. Indeed, the degree of a root of unity
of order nis given by the Euler totient function, which satisfies the lower estimate:
φ(n)≫n/ log log n(see [HW79, Thm 328]).
It follows that αhas multiplicative order at most m≤C0dlog log d. As this
is less than log plog log log pif Xis large enough (and δ0chosen small enough), a
contradiction.
Now combining Claims 2 and 3 we get:
|{p∈[√X, X]; pis δ-bad}| ≤ 10d≤X3√δ,
for all X≥1, hence
|{p≤X;pis δ-bad}| ≤ Xδ+X
k≥0;2k≤1/δ
X3√δ/2k≪δX3√δ.
This concludes the proof of Theorem 4.
3. Proof of Theorem 3
The proof that (b) implies (a) is immediate from the corresponding implication
in Theorem 4. Indeed given ǫ∈(0,1
2) and C > 1 every large enough ( 1
C, ǫ)-very
bad prime will be C-wild.
In the converse direction, in view of Theorem 4 it is enough to show that assertion
(c) of Theorem 4 implies (b) of Theorem 3. Let ǫ=1
2and let δbe given by assertion
(c). It is enough to find C=C(δ)>1 such that every C-wild prime is δ-bad. We
will show the contrapositive. So assume pis not δ-bad. Then for every x∈F×
p
with multiplicative order at least (log p)2we have
|Sd(x)| ≥ pδ
for some d≤log p. Now recall the sum-product theorem over Fp:
Theorem 8 (sum-product theorem, see 2.58 and 4.10 in [TV10]).There is ǫ > 0
such that for all primes pand all subsets A⊂Fpat least one of three possibilities
can occur: either AA +AA +AA =Fp, or |A+A|>|A|1+ǫ, or |AA|>|A|1+ǫ.
Note further that SK
d(x) + SK
d(x)⊂ S2K
d(x), while SK
d(x).SK
d(x)⊂ SdK 2
2d(x). In
particular, starting from Sd(x) and applying at most ntimes either a sumset or a
product set, we obtain a subset of Sd2n
2nd(x).
Applying the sum-product theorem, we see that after some n=n(δ) steps
S[log p]2n
2n[log p](x) must be all of Fp. Setting C= 2n+1, we conclude (by (1)) that p
is not C-wild as desired. This concludes the proof of Theorem 3.
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DPMMS, University of Cambridge, United Kingdom
E-mail address:emmanuel.breuillard@maths.cam.ac.uk
DPMMS, University of Cambridge, United Kingdom
E-mail address:peter.varju@maths.cam.ac.uk