Journal de Théorie des Nombres de Bordeaux

Henri Cohen

Computing $L$-Functions: A Survey

Antonella Perucca

The prime divisors of the number of points on abelian varieties

Régis de la Bretèche

Tim Browning

Contre-exemples au principe de Hasse pour certains tores coflasques

Hartwig Mayer

Self-intersection of the relative dualizing sheaf  on modular curves $X_1(N)$

Rafael von Känel

An effective proof of the hyperelliptic Shafarevich conjecture

Paul Fili

Paul FILI
On the heights of totally p-adic numbers
Tome 26, no 1 (2014), p. 103-109.
<http://jtnb.cedram.org/item?id=JTNB_2014__26_1_103_0>
© Société Arithmétique de Bordeaux, 2014, tous droits réservés.
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Journal de Théorie des Nombres
de Bordeaux 26 (2014), 103–109
On the heights of totally p-adic numbers
par Paul FILI
Résumé. Bombieri et Zannier ont démontré des minorations et
des majorations de la limite inférieure de la hauteur de Weil sur
le corps des nombres totalement p-adiques et sur leurs généralisa-
tions. Dans notre étude nous utilisons des techniques de la théorie
du potentiel pour généraliser les majorations de leur étude et, dans
l’hypothèse d’intégralité, améliorer un peu plus les minorations.
Abstract. Bombieri and Zannier established lower and upper
bounds for the limit infimum of the Weil height in fields of totally
p-adic numbers and generalizations thereof. In this paper, we use
potential theoretic techniques to generalize the upper bounds from
their paper and, under the assumption of integrality, to improve
slightly upon their bounds.
1. Statement of Results
Recall that an algebraic number is said to be totally p-adic if its image
lies in Qp for any embedding Q ↪→ Cp, where Cp denotes the completion
of an algebraic closure of Qp. This is analogous to the usual definition of a
totally real number, however, unlike C/R, the extension Cp/Qp is of infinite
degree, so in fact we can make an even broader generalization:
Definition 1. Let Lp/Qp be a (finite) Galois extension for p ≤ ∞ a rational
prime. We say α ∈ Q is totally Lp if all Galois conjugates of α lie in Lp ⊂ Cp.
More generally, when our objects are defined over an arbitrary number
field K, we make the following definition:
Definition 2. Fix a base number field K, and let S be a set of places of
K. For each v ∈ S, we choose a Galois extension Lv/Kv. We say that α is
totally LS/K if, for each v ∈ S, all of the K-Galois conjugates of α lie in
Lv.
Notice that α is totally LS/K if and only if the minimal polynomial for
α over K splits in Lv for each v ∈ S. In our terminology, a number being
totally real is equivalent to being totally R (or totally R/Q), and being
Manuscrit reçu le 31 octobre 2012, accepté le 11 mai 2013.
Mots clefs. Weil height, totally p-adic, potential theory, Fekete-Szegő theorem.
Classification math. 11G50, 11R06, 37P30.
104 Paul Fili
totally p-adic is equivalent to being totally Qp (totally Qp/Q). Notice that
the set of all totally LS/K numbers form a normal extension ofK (typically
of infinite degree).
Bombieri and Zannier [3] studied the question of what the limit infimum
of the Weil height was in these fields when the base field was assumed to
be Q, and they proved the following:
Theorem (Bombieri and Zannier 2001). Let S is the set of finite rational
primes and for each p ∈ S, let Lp/Qp be a chosen Galois extension. Suppose
L/Q is a normal extension (possibly of infinite degree) such that L embeds
in Lp for each p ∈ S. Then
lim inf
α∈L
h(α) ≥ 12
∑
p∈S
log p
ep(pfp + 1)
where ep and fp denote the ramification and inertial degrees of Lp/Qp,
respectively.
Further, in the case where S = {p1, . . . , pn} is a finite set of nonar-
chimedean rational primes with Lpi = Qpi for each i, if we let L be the field
of all totally LS/Q numbers, then we have
lim inf
α∈L
h(α) ≤
n∑
i=1
log pi
pi − 1
.
We note that in Bombieri and Zannier’s result, by the assumption that
our extensions are normal, it follows that every p-adic completion of L
has ramification and inertial degree less than or equal to that of Lp. More
simply, we might take L to be the field of all totally LS/Q numbers above
and recover the same result.
Smyth [8, 9] and Flammang [4] proved analogous results in the totally
real case for the limit infimum of the height, however, they imposed the
additional hypothesis that one consider only totally real algebraic integers.
This note has two aims: first, to prove a slightly sharper lower bound by
adding the hypothesis that we consider only algebraic integers, and second,
to generalize the upper bound from Bombieri and Zannier’s paper by the
aid of the Fekete-Szegő theorem with splitting conditions, as formulated
and proven by Rumely [5, 6]. Our results are the following:
Theorem 1.1. Fix a number field K, a set of nonarchimedean places S of
K, and a choice of Galois extension Lv/Kv for each v ∈ S. Let L be the
field of all totally LS/K numbers, and OL denote its ring of integers. Then
lim inf
α∈OL
h(α) ≥ 12
∑
v∈S
Nv ·
log pv
ev(qfvv − 1)
On the heights of totally p-adic numbers 105
where Nv = [Kv : Qv]/[K : Q], ev and fv denote the ramification and iner-
tial degrees of Lv/Kv, respectively, and qv denotes the order of the residue
field of Kv, and pv is rational prime above which v lies.
As Bombieri and Zannier remark in their note, if the above sum diverges
(which may happen if we allow S to be infinite), then it follows that the set
of totally LS/K algebraic integers satisfies the Northcott property (there
are only a finite number with height below any fixed constant).
If we restrict our attention to extensions normal over Q, as Bombieri and
Zannier do, our result reduces to:
Corollary 1. Fix a set of finite rational primes S and a choice of Galois
extension Lp/Qp for each p ∈ S. Let L be the field of all totally LS numbers,
and OL denote its ring of integers. Then
lim inf
α∈OL
h(α) ≥ 12
∑
p∈S
log p
ep(pfp − 1)
where ep and fp denote the ramification and inertial degrees of Lp/Qp,
respectively.
Theorem 1.2. Fix a number field K, a finite set of nonarchimedean places
S of K, and a choice of Galois extension Lv/Kv for each v ∈ S with
ramification degree ev, inertial degree fv, and residue field degree qv and
characteristic pv. Let L be the field of all totally LS/K numbers. Then
lim inf
α∈L
h(α) ≤
∑
v∈S
Nv ·
log pv
ev(qfvv − 1)
.
where Nv = [Kv : Qv]/[K : Q].
Notice that this differs from our lower bound only by the factor of 1/2.
One interesting thing to note in Theorem 1.1 is that the shape of our
bound comes directly from the formula for the logarithmic capacity of the
ring of integers of Lv at each place. This connection to potential theory
gives an interesting explanation for the shape of the observed bound, and
suggests that by determining minimal energy measures on Lv one might
further improve the bounds above. In fact, this connection inspires us to
conjecture that in fact the upper bound in Theorem 1.2 is sharp:
Conjecture 1. Fix a number field K, a finite set of nonarchimedean places
S of K, and a choice of Galois extension Lv/Kv for each v ∈ S with
ramification degree ev, inertial degree fv, and residue field of degree qv and
characteristic pv. Let L be the field of all totally LS/K numbers and OL its
ring of integers. Then
lim inf
α∈OL
h(α) =
∑
v∈S
Nv ·
log pv
ev(qfvv − 1)
.
106 Paul Fili
where Nv = [Kv : Qv]/[K : Q].
It is still an interesting open question whether the limit infimum for
all numbers can be in fact achieved with algebraic integers or not. In this
direction, we will note only that the result of Bombieri and Zannier, when
read as an equidistribution result, seems to indicate that totally p-adic
points of low height should be distributing evenly in the residue classes of
P1(Fp). This seems to suggest that the limit infimum over an entire field
L of the type constructed above might be smaller than that obtained by
algebraic integers; however, all of the smallest known limit points, both for
the height of totally real numbers (see [8]) and the height of totally p-adic
numbers (as in Theorem 1.2) are in fact achieved by sequences of algebraic
integers.
2. Proofs
Proof of Theorem 1.1. We now proceed to prove Theorem 1.1. Recall that
we have fixed a base number field K, and let MK denotes the places of K.
First, let us note that if S is infinite, we can take a limit over increasing
finite subsets of S, and the general result will follow, so we may as well
assume that S is finite in our proof. For convenience let
Nv =
[Kv : Qv]
[K : Q] .
We recall M. Baker’s reformulation of Mahler’s inequality from [1], namely,
if α is an algebraic number and α1, . . . , αn denote its K-Galois conjugates,
then
(2.1) 1
n(n− 1)
∑
i 6=j
gv(x, y) ≥
−Nv ·
logn
n− 1 if v | ∞,
0 otherwise
where
gp(x, y) = log+|x|v + log+|y|v − log|x− y|v,
and our absolute values are normalized so that |·|v = ‖·‖Nvv where ‖·‖v
extends the usual absolute of Q over which it lies (thus the absolute values
|·|v satisfy the product formula, and h(α) =
∑
v log+|α|v). We assume in
our theorem that α is integral. The key role is played by the discriminant
here. At the place v ∈MK (p here being the rational prime over which the
place v lies), all of the conjugates of α lie in ring of integers OLv . By [7,
Example 4.1.24], we have that the v-adic logarithmic capacity of OLv with
respect to the point ∞, which we will denote γ∞,v(OLv ), satisfies
log γ∞,v(OLv ) = −Nv ·
log pv
ev(qfvv − 1)
< 0,
On the heights of totally p-adic numbers 107
where qv denotes the order of the residue field of OLv and pv its charac-
teristic.1 We can also view this as the transfinite diameter of OLv ; more
specifically, if we adopt the notation of [2, Ch. 6] and let
log dn(OLv )∞ = sup
z1,...,zn∈OLv
1
n(n− 1) log|zi − zj |v,
then by [2, Lemma 6.21 and Theorem 6.23] it follows that2
lim
n→∞
log dn(OLv )∞ = log γ∞,v(OLv ).
Let {α(k)}∞k=1 denote a sequence of algebraic numbers such that
lim
k→∞
h(α(k)) = lim inf
α∈OL
h(α),
and let nk denote the number of K-Galois conjugates of α(k), which we
denote α(k)1 , . . . , α
(k)
nk . Since the α(k) have bounded height, it follows from
Northcott’s theorem that nk → ∞ as k → ∞. By definition of dn(OLv )∞,
we have
1
nk(nk − 1)
∑
i 6=j
log|α(k)i − α
(k)
j |v ≤ log dnk(OLv )∞
for any totally LS/K integer and v ∈ S, so it follows that
lim sup
k→∞
∑
v∈S
1
nk(nk − 1)
∑
i 6=j
log|α(k)i − α
(k)
j |v ≤
∑
v∈S
log γ∞,v(OLv ).
Therefore, by the product formula, using the assumption of integrality,
lim inf
k→∞
∑
w∈MK
w|∞
1
nk(nk − 1)
∑
i 6=j
log|α(k)i − α
(k)
j |w ≥ −
∑
v∈S
log γ∞,v(OLv ).
It now follows from (2.1) that
lim inf
k→∞
2h(α(k)) ≥ lim inf
k→∞
∑
w|∞
log+|α(k)|w
≥ −
∑
v∈S
log γ∞,v(OLv ) =
∑
v∈S
Nv ·
log pv
ev(qfvv − 1)
.
Upon dividing each side by 2, the result follows. 
1In the text [7] the v-adic absolute value |·|v which is used to compute the capacity is nor-
malized to agree with the modulus with respect to the additive Haar measure, so in the text a
log qv appears in the numerator instead of log pv . We also wish to draw the reader’s attention
to the fact that the ring of integers OLv is ‘strictly smaller’ than the closed unit disc in P1(Cv),
which has logarithmic capacity log γ∞,v(Dv(0, 1)) = 0.
2Note that here for convenience our absolute values are normalized so that we have the same
normalization factors as in computing the absolute logarithmic Weil height.
108 Paul Fili
Proof of Theorem 1.2. We will proceed by applying the adelic Fekete-Szegő
theorem with splitting conditions [6, Theorem 2.1] (see also [5] as well as
[2, Theorem 6.27] for an interpretation in terms of the Berkovich topology).
For each v ∈ S, let OLv ⊂ Lv denote the ring of integers in Lv. Notice that
OLv ⊂ P1(Cp) is a compact set and let Dv(0, 1) ⊂ P1(Cv) be the usual
Berkovich unit disc. Let E ⊂
∏
v∈MK A
1(Cv) be the adelic Berkovich set
given by
E =
∏
v∈MK
Ev where Ev =

OLv for v ∈ S
Dv(0, 1) for v 6∈ S,
D
(
0,
∏
v∈S
qNv/ev(q
fv
v −1)
v
)
for v | ∞.
By [7, Example 4.1.24] we have normalized v-adic logarithmic capacity
log γ∞,v(OLv ) = −Nv ·
log pv
ev(qfvv − 1)
for each v ∈ S,
where γ∞ denotes the capacity relative to X = {∞} in the notation of [6].
At the infinite places, we have∑
w|∞
log γ∞,w(D(0,
∏
v∈S
qNv/ev(q
fv
v −1)
v )) =
∑
v∈S
Nv ·
log pv
ev(qfvv − 1)
,
so that the adelic capacity, that is, the product of all of the normalized
capacities, is
γ∞(E) =
∏
v∈MK
γ∞,v(Ev) = 1.
Further, by our assumptions, E is stable under the action of the continuous
Galois automorphisms of Cv/Kv at each place. For ε > 0, we consider the
Berkovich adelic neighborhood
U =
∏
w∈MK
Uw where Uw =

Ew for w -∞
D
(
0, eε
∏
v∈S
qNv/ev(q
fv
v −1)
v
)−
for w | ∞,
where D(a, r)− denotes the usual open disc centered at a of radius r. Then
we can apply the adelic Fekete-Szegő theorem with splitting conditions as
formulated in [6, Theorem 2.1], there are infinitely many algebraic numbers
in K all of whose conjugates lie in U with the additional condition that all
Cv/Kv conjugates lie in Uv = Ev = OLv for each v ∈ S. Clearly, the only
contribution to the height of such numbers comes from the archimedean
places, and as a result, we have
h(α) ≤
∑
v∈S
Nv ·
log pv
ev(qfvv − 1)
+ ε.
On the heights of totally p-adic numbers 109
Thus by letting ε → 0 we can generate a sequence of numbers (in fact,
algebraic integers) with the limit infinimum of the height bounded by the
desired constant. 
References
[1] M. Baker, A lower bound for average values of dynamical Green’s functions, Math. Res.
Lett., 13 (2006), 245–257.
[2] M. Baker and R. Rumely, Potential theory and dynamics on the Berkovich projective line,
Mathematical Surveys and Monographs 159, American Mathematical Society, Providence,
RI, (2010).
[3] E. Bombieri and U. Zannier, A note on heights in certain infinite extensions of Q, Atti
Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001), 5–14.
[4] V. Flammang, Two new points in the spectrum of the absolute Mahler measure of totally
positive algebraic integers, Math. Comp. 65 (1996), 307–311.
[5] R. Rumely, The Fekete-Szegő theorem with splitting conditions. I, Acta Arith. 93 (2000), 99–
116.
[6] R. Rumely, The Fekete-Szegő theorem with splitting conditions. II, Acta Arith. 103
(2002), 347–410.
[7] R. Rumely, Capacity theory on algebraic curves, Lecture Notes in Mathematics 1378,
Springer-Verlag, Berlin, (1989).
[8] C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A
30 (1980/81), 137–149.
[9] C. J. Smyth, On the measure of totally real algebraic integers. II, Math. Comp. 37
(1981), 205–208.
Paul Fili
Department of Mathematics
University of Rochester, Rochester, NY 14627
E-mail: fili@post.harvard.edu


On the heights of totally $p$-adic numbers

T. Alden Gassert

Discriminants of Chebyshev radical extensions

Cécile Le Rudulier

Points algébriques de hauteur bornée  sur la droite projective

Iurie Boreico

Daniel El-Baz

Thomas Stoll

Iurie BOREICO, Daniel EL-BAZ et Thomas STOLL
On a conjecture of Dekking : The sum of digits of even numbers
Tome 26, no 1 (2014), p. 17-24.
<http://jtnb.cedram.org/item?id=JTNB_2014__26_1_17_0>
© Société Arithmétique de Bordeaux, 2014, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 26 (2014), 17–24
On a conjecture of Dekking : The sum of digits
of even numbers
par Iurie BOREICO, Daniel EL-BAZ et Thomas STOLL
Résumé. A propos d’une conjecture de Dekking : la somme des
chiffres des nombres pairs
Soient q ≥ 2 et sq la fonction somme des chiffres en base q. Pour
j = 0, 1, . . . , q − 1 on considère
#{0 ≤ n < N : sq(2n) ≡ j (mod q)}.
En 1983, F. M. Dekking a conjecturé que cette quantité est stric-
tement supérieure à N/q et, respectivement, strictement inférieure
à N/q pour une infinité de N , affirmant ce faisant l’absence d’un
phénomène de dérive (ou phénomène de Newman). Dans cet ar-
ticle, nous démontrons sa conjecture.
Abstract. Let q ≥ 2 and denote by sq the sum-of-digits func-
tion in base q. For j = 0, 1, . . . , q − 1 consider
#{0 ≤ n < N : sq(2n) ≡ j (mod q)}.
In 1983, F. M. Dekking conjectured that this quantity is greater
than N/q and, respectively, less than N/q for infinitely many N ,
thereby claiming an absence of a drift (or Newman) phenomenon.
In this paper we prove his conjecture.
1. Introduction
Let q ≥ 2 and denote by sq : N → N the sum-of-digits function in the
q-ary digital representation of integers. In his influential paper from 1968,
Gelfond [5] proved the following result.1
Theorem 1.1. Let q, d,m ≥ 2 and a, j be integers with 0 ≤ a < d and
0 ≤ j < m. If (m, q − 1) = 1 then
(1.1)
#{ 0 ≤ n < N : n ≡ a (mod d), sq(n) ≡ j (mod m)} =
N
dm
+ g(N),
where g(N) = Oq(Nλ) with λ = 12 log q log
q sin(π/2m)
sin(π/2mq) < 1.
Manuscrit reçu le 18 novembre 2012, accepté le 19 février 2013.
1As usual, we write f(N) = O(1) if |f(N)| < C for some absolute constant C, and f(N) =
Oq(1) if the implied constant depends on q.
18 Iurie Boreico, Daniel El-Baz, Thomas Stoll
Shevelev [8, 9] recently determined the optimal exponent λ in the error
term in Gelfond’s asymptotic formula when q = m = 2, and Shparlinski [10]
showed that in this case it can be arbitrarily small for sufficiently large
primes d.
The oscillatory behaviour of the error term g(N) in (1.1) is still not
completely understood. The story can be said to have originated with the
observation by Moser in the 1960s that for the quintuple of parameters
(1.2) (q, a, d, j,m) ≡ (2, 0, 3, 0, 2)
the error term seems to have constant positive sign, i.e., g(N) > 0 for all
N ≥ 1. In 1969, Newman [7] (with a much more precise result by Coquet [2])
proved this observation and there is at present a large number of articles
which establish so-called Newman phenomena, Newman-like phenomena or
drifting phenomena for general classes of quintuples (q, a, d, j,m) extend-
ing (1.2). The two main techniques come from a direct inspection of the
recurrence relations using the q-additivity of the sum-of-digits function,
and from the determination of the maximal and minimal value of a related
fractal function which is continuous but nowhere differentiable [6, 2, 11].
We refer the reader to the monograph of Allouche and Shallit [1] and the
article of Drmota and Stoll [4] for a list of references. Characterizing all
(q, a, d, j,m) for which one has a Newman-like phenomenon is still wide
open.
The aim of the present article is to prove a related conjecture Dekking
(see [3, “Final Remark”, p. 32-11]) made in 1983 at the Séminaire de Théorie
des Nombres de Bordeaux concerning a non-drifting phenomenon, that is,
a situation where the error g(N) is oscillating in sign (as N →∞). To our
knowledge, this conjecture has not yet been addressed in the literature, and
we will provide a self-contained proof here.
Conjecture (Dekking, 1983): Let q ≥ 2 and 0 ≤ j < q and set
(q, a, d, j,m) ≡ (q, 0, 2, j, q).
Then g(N) < 0 and g(N) > 0 infinitely often.
Dekking was mostly interested in finding the optimal error term in (1.1)
(or, as he puts it, the typical exponent of the error term) and obtained
various results for the cases q = 2, d arbitrary, and d = 2, q arbitrary. As
for the conjecture, he proved the case of q = 3, j = 0, 1, 2 via an argument
with a geometrical flavour.
Our main result is as follows.
Theorem 1.2. Let q ≥ 2, 0 ≤ j < q and set
(q, a, d, j,m) ≡ (q, a, d, j, q).
On a conjecture of Dekking 19
(i) If d | q, then g(N) = O(1) and g(N) changes signs infinitely often
as N →∞.
(ii) If d | q − 1, then g(N) can take arbitrarily large positive values as
well as arbitrarily large negative values as N →∞.
In the case of d = 2 this proves Dekking’s conjecture and covers all bases
q ≥ 2.
2. Proof of Theorem 1.2
For an integer n ≥ 0, we write
n = (εk, εk−1, . . . , ε0)q
to refer to its q-ary digital expansion n =
∑k
i=0 εiq
i. Let U(n) = {z ∈
C | zn = 1} denote the set of the nth roots of unity. We will make use of
the following well-known formula from discrete Fourier analysis.
Proposition 2.1. Let f(x) =
∑∞
k=0 akx
k ∈ C[x], n ≥ 1, l ≥ 0 and set
ωn = e2πi/n. Then ∑
k≡l ( mod n)
akx
k = 1
n
n−1∑
s=0
ω−lsn f(ωsnx).
Proof. The coefficient of xj in 1n
∑n−1
s=0 ω
−ls
n f(ωsnx) is 1n
∑n−1
s=0 ajω
s(j−l)
n , that
is aj if j ≡ l (mod n) and 0 otherwise. 
We deal with (i) d | q and (ii) d | q − 1 in Theorem 1.2 separately in the
two subsequent sections.
2.1. The case d | q. For d = 2, q even, Dekking remarked and left to the
readers of his article (see [3, Remark before Proposition 5, p.32-08]) that
the typical exponent λ equals 0, i.e., g(N) = O(1). This is due to the fact
that when q is even then the parity of an integer is completely encoded
in the last digit of its base q expansion. A similar situation applies when
d | q. In order to find the oscillatory behaviour of g(N), we calculate g(N)
explicitly.
Define
fj(n) = cj(n)−
1
q
,
where
cj(n) =
{
1 if sq(n) ≡ j (mod q);
0 otherwise.
Consider
(2.1) Dj(N) =
∑
0≤n<N
n≡a (mod d)
fj(n),
20 Iurie Boreico, Daniel El-Baz, Thomas Stoll
thus
(2.2) g(N) = Dj(N)−
N
dq
+ 1
q
⌈
N − a
d
⌉
.
We want to find infinitely many values of N such that g(N) > 0, respec-
tively, g(N) < 0. Since an integer in base q (with q divisible by d) gives
remainder a mod d if and only if its last digit in base q gives remainder a
mod d, we get for N = (εk, . . . , ε0)q,
Dj(N) =
k∑
r=2
εr−1∑
δ=0
∑
0≤i0,i1,...,ir−1≤q−1
i0≡a (mod d)
fj((εk, . . . , εr+1, δ, ir−1, . . . , i0)q)
+
ε1−1∑
δ=0
q−1∑
i0=0
i0≡a (mod d)
fj((εk, . . . , ε2, δ, i0)q)
+
ε0−1∑
δ=0
δ≡a (mod d)
fj((εk, . . . , ε1, δ)q).
For r ≥ 2 we get ∑
0≤i0,i1,...,ir−1≤q−1
i0≡a (mod d)
fj((εk, . . . , εr+1, δ, ir−1, . . . , i0)q)
= Dj−εk−···−εr+1−δ(q
r) = 0.
Set α = j − sq(N) + ε1 + ε0 and β = j − sq(N) + ε0. For the other two
terms we then get by a direct calculation,
(2.3)
ε1−1∑
δ=0
q−1∑
i0=0
i0≡a (mod d)
fj((εk, . . . , ε2, δ, i0)q) = −
ε1
d
+
ε1−1∑
δ=0
∑
0≤i0<q
i0≡a (mod d)
i0≡α−δ (mod q)
1
and
(2.4)
ε0−1∑
δ=0
δ≡a (mod d)
fj((εk, . . . , ε1, δ)q) = −
1
q
⌈
ε0 − a
d
⌉
+
∑
0≤δ<ε0
δ≡a (mod d)
δ≡β (mod q)
1.
From (2.2), (2.3) and (2.4) it is straightforward to find sequences of positive
integers N with g(N) > 0, respectively g(N) < 0. In fact, if a 6= 0 we can
take all N with ε1 = 0, ε0 = a to get g(N) = − aqd < 0. For a = 0 we take all
N with ε1 = 1, ε0 = a and sq(N) 6≡ j+1 (mod d) to get g(N) = −1/d < 0.
On the other hand, if a+ 1 < q we may take all N with ε1 = 0, ε0 = a+ 1
to find g(N) = 1 + 1d −
a+1
qd −
1
q > 0. If a + 1 = q (which again implies
On a conjecture of Dekking 21
d = q) we take all N with ε1 = 1, ε0 = 0 and sq(N) ≡ j+ 2 (mod q) to get
g(N) = −1d + 1 > 0. This completes the proof in this case.
2.2. The case d | q − 1. In what follows, set
Ea,j(k) = #{ 0 ≤ n < qk : n ≡ a (mod d), sq(n) ≡ j (mod q)},
where a, j are fixed integers with 0 ≤ a < d, 0 ≤ j < q and k ≥ 1. Consider
the generating polynomial in two variables
P (x, y) =
k−1∏
i=0
(1 + xyqi + x2y2qi + · · ·+ xq−1y(q−1)qi),
which encodes the digits of integers less than qk in base q. Denote by
[xuyv]P (x, y) the coefficient of xuyv in the expansion of P (x, y). By Propo-
sition 2.1,
Ea,j(k) =
∑
u≡j (mod q)
v≡a (mod d)
[xuyv]P (x, y) = 1
dq
∑
ω∈U(q)
ε∈U(d)
ω−jε−aP (ω, ε).
For ε ∈ U(d) with d | q − 1 we have εlqi = εl for 0 ≤ l ≤ q − 1 and thus
P (ω, ε) = (1 + ωε+ ω2ε2 + . . .+ ωq−1εq−1)k.
Since ωε = 1 if and only if ω = ε = 1 (d and q are coprime) and ωqεq = ε
we get
(2.5) Ea,j(k)−
qk−1
d
= 1
dq
∑
ω∈U(q)
ε∈U(d)
ωε6=1
ω−jε−a
( 1− ε
1− ωε
)k
.
We now take a closer look at the dominant term on the right hand side
in (2.5). Note that for ω ∈ U(q), ε ∈ U(d) with ωε 6= 1, we have
1
π
arg
( 1− ε
1− ωε
)
∈ Q.
We claim that the numbers 1− ε1− ωε are all pairwise distinct. Indeed, for
any point on the unit circle z 6= 1, it can easily be seen (geometrically or
otherwise) that arg((1− z)2) = arg(z) + π. It follows that
arg
(( 1− ε
1− ωε
)2)
= − arg(ω).
Therefore, if
1− ε
1− ωε =
1− ε′
1− ω′ε′
22 Iurie Boreico, Daniel El-Baz, Thomas Stoll
then we conclude that ω and ω′ have the same argument so ω = ω′, and
then ε = ε′. This means that there are no cancellations in (2.5).
Write
R = max
{∣∣∣∣ 1− ε1− ωε
∣∣∣∣ : ω ∈ U(q), ε ∈ U(d), ωε 6= 1}
and let r1, r2, . . . , rh be all of the numbers (1− ε)/(1− ωε) whose absolute
value equals R.
The set U(d) divides the unit circle into d ≥ 2 equal parts, so it always
contains an element ε0 in the open half-plane Re(ε) < 0. Similarly, U(q)
must contain an element ω0 in the closed half-plane Re(ε0ω) ≥ 0. Then
|1− ε0| >
√
2 while |1− ω0ε0| ≤
√
2, thus∣∣∣∣ 1− ε01− ω0ε0
∣∣∣∣ > 1.
Note also that ω0ε0 6= 1 as (d, q) = 1 and ε0 6= 1.
It follows that R > 1, which in particular implies that the value 1 is not
among these ri. Then, as k →∞,∑
ω∈U(q)
ε∈U(d)
ωε6=1
ω−jε−a
( 1− ε
1− ωε
)k
∼ Rk
h∑
i=1
ci
(
ri
R
)k
,
for certain ci ∈ C which are not all zero. As the ri all have arguments
equal to rational multiples of π, the ri/R, i = 1, . . . , h, are roots of unity.
Therefore there exists an integer M ≥ 1 such that (ri/R)M = 1 for all i.
Write
c′(k) =
h∑
i=1
ci
(
ri
R
)k
.
Since Ea,j(k) is real and c′(k + M) = c′(k) for all k we must have that
c′(k) ∈ R for all k. Moreover,
M−1∑
k=0
c′(k) =
h∑
i=1
ci
M−1∑
k=0
(
ri
R
)k
= 0,
since ri is not real for all i. Thus, among all the c′(k) there is at least
one positive and at least one negative value. Let −c′1 = c′(k1) < 0 be the
smallest negative value and c2 = c′(k2) > 0 be the largest positive value
among them. Then, as k →∞,
Ea,j(k)−
qk−1
d
∼ − c
′
1
dq
Rk < 0, for k ≡ k1 (mod M)
and
Ea,j(k)−
qk−1
d
∼ c
′
2
dq
Rk > 0, for k ≡ k2 (mod M).
This completes the proof.
On a conjecture of Dekking 23
3. Acknowledgment
We would like to express our gratitude to J.-P. Allouche, F. M. Dekking
and J. Shallit for related correspondence on this problem. T. Stoll was
supported by the Agence Nationale de la Recherche, grant ANR-10-BLAN
0103 MUNUM.
References
[1] J.-P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations.
Cambridge University Press, (2003).
[2] J. Coquet, A summation formula related to the binary digits. Invent. Math. 73 (1983),
107–115.
[3] F. M. Dekking, On the distribution of digits in arithmetic sequences. Séminaire de Théorie
des Nombres de Bordeaux, exposé no.32 (1983).
[4] M. Drmota, T. Stoll, Newman’s phenomenon for generalized Thue-Morse sequences, Dis-
crete Math. 308, (7) (2008), 1191–1208.
[5] A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données,
Acta Arith. 13 (1968), 259–265.
[6] S. Goldstein, K. A. Kelly, E. R. Speer, The fractal structure of rarefied sums of the
Thue-Morse sequence, J. Number Theory 42 (1992), 1–19.
[7] D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math.
Soc. 21 (1969), 719–721.
[8] V. Shevelev, Generalized Newman phenomena and digit conjectures on primes, Int. J.
Math. Math. Sci., ID 908045 (2008).
[9] V. Shevelev, Exact exponent in the remainder term of Gelfond’s digit theorem in the binary
case, Acta Arith. 136 (2009), 91–100.
[10] I. Shparlinski, On the size of the Gelfond exponent, J. Number Theory 130, (4) (2010),
1056–1060.
[11] G. Tenenbaum, Sur la non-dérivabilité de fonctions périodiques associées à certaines fonc-
tions sommatoires, in: R.L. Graham & J. Nesetril (eds), The mathematics of Paul Erdős,
Algorithms and Combinatorics 13 Springer Verlag, (1997), 117–128.
24 Iurie Boreico, Daniel El-Baz, Thomas Stoll
Iurie Boreico
Department of Mathematics
Stanford University
450 Serra Mall
Stanford, California 94305, USA
E-mail: boreico@math.stanford.edu
Daniel El-Baz
School of Mathematics
University of Bristol
University Walk
Bristol, BS8 1TW, United Kingdom
E-mail: Daniel.El-Baz@bristol.ac.uk
Thomas Stoll
1. Université de Lorraine
Institut Elie Cartan de Lorraine, UMR 7502
Vandoeuvre-lès-Nancy, F-54506, France
2. CNRS
Institut Elie Cartan de Lorraine, UMR 7502
Vandoeuvre-lès-Nancy, F-54506, France
E-mail: thomas.stoll@univ-lorraine.fr


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