Polynomial values and generators
with missing digits in finite fields
Cécile Dartyge (Nancy), Christian Mauduit (Marseille) and András Sárközy (Budapest)
Abstract. We consider the linear vector space formed by the elements of the finite
field Fq with q =Ppr over Fp . Then the elements x of Fq have a unique representation
r
in the form x = j=1 cj aj with cj ∈ Fp ; the coefficients cj will be called digits. Let D
be a subset of Fp with 2 6 |D| < p. We consider elements x of Fq such that for their
every digit cj we have cj ∈ D; then we say that the elements of Fp r D are “missing
digits”. We will show that if D is a large enough subset of Fp , then there are squares
with missing digits in Fq ; if the degree of the polynomial f (x) ∈ Fq [X] is at least 2
then it assumes values with missing digits; there are generators g in Fq such that f (g)
is of missing digits.
1. Introduction
Let b ∈ N be fixed with b > 2. If n ∈ N, then consider the representation of n in the
number system to base b:
(1·1)
n=
r−1
X
j=0
cj bj ,
0 6 cj 6 b − 1, cr−1 > 1,
and write
(1·2)
S(n) =
r−1
X
cj .
j=0
Many papers have been written on the connection between the arithmetic properties
of n and certain properties of its digits c0 , c1 , . . . cr−1 . In particular the sum of digits
function S(n) restricted to polynomial or prime numbers has been studied in [12], [13],
[15-17], [20-22], [27], [28], [30], [31], [32]. In some other papers [1-9], [14], [18], [19], [24],
[25], [29] the arithmetic properties of integers with missing digits have been studied.
In [11] Dartyge and Sárközy initiated the study of the analogs of some of these
problems in finite fields. Indeed, let p be a prime number, q = pr with r > 2, and
consider the field Fq . Let B = {a1 , a2 , . . . , ar } be a basis of the linear vector space
formed by Fq over Fp , i. e., let a1 , a2 , . . . , ar be linearly independent over Fp . Then
every x ∈ Fq has a unique representation in form
(1·3)
x=
r
X
cj aj
j=1
2010 Mathematics Subject Classification: Primary 11A63; Secondary 11L99.
Key words and phrases: digits properties, finite fields, character sums, squares, polynomials,
generators, primitive roots.
∗
Research partially supported by the Hungarian National Foundation for Scientific Research,
Grant K72731 and K100291 and the Agence Nationale de la Recherche, grant ANR-10-BLAN 0103
MUNUM and French-Hungarian “Balaton” exchange program TÉT-09-1-2010-0056.
∗
2
with cj ∈ Fp . Write
(1·4)
r
X
SB (x) =
cj .
j=1
An important special case is when the basis B consists of the first r powers of a
generator z of F∗q :
B = {a1 , a2 , . . . , ar } = {1, z, z 2 , . . . , z r−1 }.
Then (1·3) becomes
(1·5)
x=
r
X
cj z j .
j=1
(1·4) and (1·5) are of the same form as (1·1) and (1·2), thus we may consider (1·3) as
the finite field analog of the representation (1·1), and we may call c1 , c2 , . . . , cr in (1·3)
as “digits”, and SB (x) can be called as “sum of digits ” function. It was shown in [11]
that if we fix an s ∈ Fp and f (x) ∈ Fq [x] satisfying certain assumptions then there are
squares x2 , elements y ∈ Fq and generators g ∈ Fq with SB (x2 ) = s, SB (f (y)) = s,
SB (f (g)) = s respectively.
In this paper our goal is to prove similar results for the elements x ∈ Fq with missing
digits. More precisely, let us fix a set D ⊂ {0, 1, 2, . . . , p − 1} with 2 6 |D| 6 p − 1. We
define the set WD as the set of all elements x ∈ Fq such that all their digits belong to
D in the basis B = {a1 , a2 , . . . , ar }:
r
n
o
X
WD = x =
cj aj with (c1 , . . . , cr ) ∈ Dr .
j=1
Then we have |WD | = |D|r , and the elements of {0, 1, . . . , p − 1} r D are called missing
digits. Let Q denote the set of the quadratic residues of Fq and for f (x) ∈ Fq [x], WD (f )
the set of the polynomial values of f (x) with missing digits:
WD (f ) = {x ∈ Fq : f (x) ∈ WD }.
In order to formulate some of our results we will also use the notation
(1·6)
C(p, t) =
(
≥
¥
log p
log 3
1 4
+ p1
t + t 3 − 2
2
2
π
p + π(p−1) (1 − log(2 sin 2p ))
if 2 6 t < p − 2,
if t = p − 2.
We will show that if D is a large subset of Fp , then there are many squares in Q with
missing digits; if the degree of the polynomial f (x) ∈ Fq [x] is at least 2 then it assumes
values with missing digits; there are generators g in Fq such that f (g) is of missing
digits. We remark that the analog problems in N (on squares, polynomial values and
primes with missing digits) are open and seem to be very difficult.
Polynomial values and generators with missing digits in finite fields
2. Squares with missing digits
First we prove that if |D| is close to p, then half of the elements of WD are quadratic
residues.
Theorem 2.1. Let D ⊂ Fp with 2 6 |D| 6 p − 1. Then we have
(2·1)
Ø
¥r
p
|WD | ØØ
1 ≥
Ø
|W
∩
Q|
−
6
|D|
+
p
p
−
|D|
.
Ø D
Ø
√
2
2 q
Remark. Theorem 2.1 gives a non-trivial upper bound if
p
√
|D| + p p − |D| < |D| p,
that is
|D| >
(2·2)
−p2 +
p
√
p4 + 4p3 ( p − 1)2
√
2( p − 1)2
√
( 5 − 1)p
∼
2
(p → +∞).
Proof.
The first step of the proof of Theorem 2.1 is to prove the following lemma. We will
use the standard notation e(t) = exp(2iπt).
Lemma 2.2. We have
p−1
Ø
|WD | ØØ
1 ≥ X ØØ X ≥ ch ¥ØØ¥r
Ø
e
Ø|WD ∩ Q| −
Ø6 √
Ø
Ø .
2
2 q
p
h=0 c∈D
Let γ denote the quadratic character of Fq . Then we have
(2·3)
|WD ∩ Q| =
1 X
|WD | 1 X
(1 + γ(x)) =
+
γ(x).
2
2
2
x∈WD
x∈WD
Next following [11], we switch to additive characters by using Gaussian sums in order
to separate the digits c1 , . . . , cr . We recall that if χ is a multiplicative character of F∗q
and ψ is an additive character of Fq then the Gaussian sum of χ and ψ is defined by
(2·4)
G(χ, ψ) =
X
χ(x)ψ(x)
x∈F∗
q
(see [26]). Then we use the following formula for all x ∈ F∗q :
χ(x) =
1X
G(χ, ψ)ψ(x).
q
ψ
Inserting this in (2·3) we obtain:
|WD ∩ Q| =
|WD |
1 X
+
G(γ, ψ)S(ψ)
2
2q
ψ
3
4
with
X
S(ψ) =
ψ
c1 ,...,cr ∈D
r
≥X
i=1
r ≥X
¥ Y
¥
ci ai =
ψ(cai ) .
i=1
c∈D
√
If ψ and χ are not both trivial, then |G(χ, ψ)| 6 q.
°k¢
For all ψ and ai , ψ(ai ) ∈ {e p , 0 6 k < p − 1}. There is a correspondence between
the additive characters ψ and Frp . This correspondence is given by (ψ(a1 ), . . . , ψ(ar )) =
≥ ° ¢
° ¢¥
e kp1 , . . . , e kpr . Thus we have
(2·5)
Ø
|WD | ØØ
1
Ø
Ø|WD ∩ Q| −
Ø6 √
2
2 q
X
06h1 ,...,hr <p
(h1 ,...,hr )6=(0,...0)
r ØX ≥
Y
hi c ¥ØØ
Ø
e
Ø
Ø.
p
i=1
c∈D
This ends the proof of Lemma 2.2.
The second part of the proof of Theorem 2.1 is to obtain an upper bound for
X ≥ hc ¥
e
.
p
c∈D
When D is large we can use the following very simple fact for h 6= 0 :
X ≥ hc ¥
X ≥ hc ¥
e −
=−
e −
,
p
p
c∈D
c∈D
with the notation D = Fp r D. This remark gives something non-trivial if |D| < |D|.
The main tool for the upper bound is the following result of Vinogradov (Lemma
14a in [33], Chapter VI page 128):
Lemma 2.3(Vinogradov’s Lemma). If α(x) and β(x) are complex valued functions
on {0, . . . , m − 1} and a 6∈ mZ then we have
m−1
Ø m−1
≥ axy ¥Ø
ØX X
Ø
α(x)β(y)e
Ø
Ø 6 (XY m)1/2 .
m
x=0 y=0
with
X=
m−1
X
x=0
|α(x)|2 and Y =
m−1
X
y=0
|β(y)|2 .
Recently Gyarmati and the third author [23] obtained a generalization of this lemma
in finite fields.
For 0 < x < p we define
≥
¥
P
xd
e
−
X ≥ xd ¥
d∈D
p
≥
¥Ø if
α(x) = ØØ P
e −
6= 0,
Ø
p
e − xd Ø
Ø
d∈D
p
d∈D
and α(x) = 1 otherwise. With this notation we have:
p−1 Ø X ≥
p−1 p
≥ xy ¥
X
xc ¥ØØ X X
Ø
e
α(x)1D (y)e
.
Ø
Ø=
p
p
x=1
x=1 y=1
c∈D
We can now apply Lemma 2.3:
p−1 Ø X ≥
X
p
p
ch ¥ØØ
Ø
(2·6)
e
Ø
Ø 6 |D| + p(p − 1)(p − |D|) 6 |D| + p p − |D|.
p
h=0 c∈D
Then we insert this bound in Lemma 2.2 and obtain the estimate (2·1) asserted in
Theorem 2.1.
5
Polynomial values and generators with missing digits in finite fields
3. Squares with missing digits when D consists of consecutive
integers
Now we will prove that if D is a set of consecutive integers with at least ¿
elements then a similar conclusion holds as in Theorem 2.1:
√
p log p
Theorem 3.1. We suppose that D = {0, . . . , t − 1} with 2 6 t 6 p − 1. Then we have:
Ø
|WD | ØØ 1
√
Ø
(3·1)
Ø|WD ∩ Q| −
Ø 6 (C(p, t)t p)r .
2
2
√
Remark. Theorem 3.1 is non-trivial if C(p, t) p < 1.
Proof. When D is a set of consecutive integers, we can apply some results of the two
first authors.
Lemma 3.2([9] Lemma 3.1). (i) If D = {0, . . . , t − 1} then
p−1
1 X ØØ X ≥ hc ¥ØØ log p 1 ≥ 4 log 3 ¥ 1
e
+
−
+ .
Ø
Ø6
pt
p
t
t 3
2
p
h=0 c∈D
(ii) If D = {0, . . . , p − 2} (and p > 3) then
p−1
1 X ØØ X ≥ hc ¥ØØ 2
2
π
e
(1 − log(2 sin )).
Ø
Ø6 +
pt
p
p π(p − 1)
2p
h=0 c∈D
Then Theorem 3.1 can be proved by combining this lemma with Lemma 2.2.
4. Polynomial values with missing digits
We obtain a similar result for |WD (f )| as the estimates in Sections 2 and 3 (but now
we will also need Weil’s theorem to achieve this). We begin by stating the result for
large |D|.
Theorem 4.1. Let D ⊂ Fp with 2 6 |D| 6 p − 1 and f (x) ∈ Z[x] with degree n > 2.
Then we have
(4·1)
Ø
Ø n − 1≥
¥r
p
Ø
Ø
|D| + p p − |D| .
Ø|WD (f )| − |WD |Ø 6 √
q
When D is a set of consecutive integers we will prove:
Theorem 4.2. We suppose that D = {0, . . . , t − 1} with 2 6 t 6 p − 1. Then we have:
Ø
Ø
√
Ø
Ø
Ø|WD (f )| − |WD |Ø 6 (n − 1)(C(p, t)t p)r .
Proofs of Theorem 4.1 and Theorem 4.2. First, we obtain in the following lemma, a
similar result as Lemma 2.2 for the sets WD (f ).
Lemma 4.3. We suppose that the degree of f is > 2. Then we have:
Ø
Ø (n − 1)
Ø
rØ
|W
(f
)|
−
|D|
Ø D
Ø6 √
q
X
06h1 ,...,hr <p
(h1 ,...,hr )6=(0,...,0)
r ØX ≥
Y
hc ¥ØØ
Ø
e
−
Ø
Ø.
p
i=1
c∈D
6
For 1 6 j 6 r we consider the additive character ψj defined by:
ψj (ai ) =
Thus for x =
Pr
i=1
Ω
exp
1
°1¢
p
if i = j
otherwise
(1 6 i 6 r).
xi ai ∈ Fq we have
≥x ¥
j
ψj (x) = ψj (x1 a1 ) · · · ψj (xr ar ) = ψjx1 (a1 ) · · · ψjxr (ar ) = e
.
p
We can use this to detect digit conditions since for x = x1 a1 + · · · + xr ar we have
p
≥ hc ¥ n
1X h
1 if xj = c
ψj (x)e −
=
p
p
0 otherwise.
h=1
If we sum this formula over all c ∈ D, then we detect the x such that xj ∈ D. It remains
then to take the product over all the digits to have an indicator of the elements of WD :
p−1
r ≥
Y
1 X X
j=1
p
c∈∈D h=0
≥ hc ¥¥ n
1 if x ∈ WD
ψjh (x)e −
=
p
0 otherwise.
We deduce that
|WD (f )| =
p
r
≥ hc ¥
XY
1 XX h
ψj (f (x))e −
.
p
p
j=1
x∈Fq
c∈D h=1
We develop this product and change the order of summation. The contribution of
h1 = . . . = hr = 0 provides the main term:
1
|WD (f )| = |D| + r
p
r
X
r
X ≥Y
x∈Fq
06h1 ,...,hr <p
(h1 ,...,hr )6=(0,...,0)
r ≥X ≥
¥Y
hc ¥¥
e −
.
p
i=1
ψihi (f (x))
i=1
c∈D
Qr
We can check easily that if (h1 , . . . , hr ) 6= (0, . . . , 0) then i=1 ψihi 6= ψ0 . Thus we can
apply the following theorem ([34], see also [26] Theorem 5.38 p. 223):
Theorem 4.4(Weil). Let g ∈ Fq [X] be of degree n > 1 with (n, q) = 1 and ψ a
nontrivial additive character of Fq . Then
ØX
Ø
√
Ø
Ø
ψ(g(x))
Ø
Ø 6 (n − 1) q.
x∈Fq
It remains to apply this theorem to finish the proof of Lemma 4.3. (4·1) is obtained
by combining Lemma 4.3 with (2·6). It is also sufficient to combine Lemma 4.3 with
Lemma 3.2 to end the proof of Theorem 4.2.
Polynomial values and generators with missing digits in finite fields
7
5. Polynomial values with generator argument and missing digits
Another variant of these problems is to study polynomial values with generator
argument. According to the notations of [11] we will denote the set of the generators
(or primitive elements) of Fq by G. For f (x) ∈ Fq [x] we now consider
WD (f, G) = {g ∈ G : f (g) ∈ WD }.
Combining the method of the proof of the previous theorem with the estimates of
character sums over generators and with polynomial arguments obtained in [11] we
can prove:
Theorem 5.1. Let D ⊂ Fp with 2 6 |D| 6 p − 1 and f (x) ∈ Z[x] with degree n > 2.
Then we have
Ø
¥r
p
ϕ(q − 1) ØØ ≥ 1 (n − 1)τ (q − 1) ¥≥
Ø
+
|D| + p p − |D|
Ø|WD (f )| − |D|r
Ø6
√
q
q
q
When D is a set of consecutive integers, the corresponding result is
Theorem 5.2. We suppose that D = {0, . . . , t − 1} with 2 6 t 6 p − 1. Then we have:
Ø
Ø
√
Ø
r ϕ(q − 1) Ø
Ø|WD (f )| − |D|
Ø 6 (1 + (n − 1)τ (q − 1))(C(p, t)t p)r .
q
The proof of Lemma 4.3 can be adapted to detect the polynomial values with
generator arguments and missing digits:
|WD (f, G)| =
p
r
≥ hc ¥
XY
1 XX h
ψj (f (g))e −
.
p
p
j=1
g∈G
c∈D h=1
We argue in the same way as before. The only difference is that instead of applying
Theorem 4.4 we use the following lemma proved in [11].
Lemma 5.3([11] Lemma 4.1). Under the notations and hypothesis of Theorem 4.4
we have:
ØX
Ø
ϕ(q − 1)
√
Ø
Ø
ψ(f (g))Ø 6 (n − 1)τ (q − 1) q +
.
Ø
q−1
g∈G
The analogue of Lemma 4.3 is then
(5·1)
p
Ø
ϕ(q − 1) ØØ ≥ 1 (n − 1)τ (q − 1) ¥≥ X ØØ X ≥ hc ¥ØØ¥r
Ø
+
e −
Ø|WD (f, G)| − |D|r
Ø6
Ø
Ø .
√
q
q
q
p
h=1 d∈D
Finally Theorem 5.1 is obtained by (5·1) and (2·6), Theorem 5.2 is proved by using
(5·1) and Lemma 3.2.
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Cécile Dartyge
Institut Élie Cartan
UMR 7502
Université de Lorraine,
BP 239
54506 Vandœuvre Cedex, France
[email protected]
Chritian Mauduit
Université Aix-Marseille
Institut de Mathématiques de Luminy
CNRS, CMR 6206
163 avenue de Luminy,
13288 Marseille cedex 9, France
[email protected]
András Sárközy
Department of Algebra
and Number Theory
Eötvös Loránd University
1117 Budapest,
Pázmány Péter sétány 1/C
Hungary
[email protected]