Double Character Sums over Subgroups and Intervals

0.2 MB·

by  Mei-Chu Chang

DOUBLE CHARACTER SUMS OVER SUBGROUPS
AND INTERVALS
4
1 MEI-CHU CHANG AND IGOR E. SHPARLINSKI
0
2
Abstract. We estimate double sums
b
e S (a, , )= χ(x+aλ), 1 a<p 1,
χ
F I G ≤ −
x∈Iλ∈G
XX
5 with a multiplicative character χ modulo p where = 1,...,H
1 I { }
and is a subgroup of order T of the multiplicative group of the
G
finite field of p elements. A nontrivial upper bound on S (a, , )
] χ I G
T can be derived from the Burgess bound if H p1/4+ε and from
N some standard elementary arguments if T p≥1/2+ε, where ε > 0
≥
. is arbitrary. We obtain a nontrivial estimate in a wider range of
h
parameters H and T. We also estimate double sums
t
a
m Tχ(a, )= χ(a+λ+µ), 1 a<p 1,
G ≤ −
[ λX,µ∈G
andgiveanapplicationtoprimitiverootsmodulopwith3non-zero
2
binary digits.
v
1
1
6
6
. 1. Introduction
1
0
1.1. Background and motivation. For a prime p, we use F to de-
4 p
1 note the finite field of p elements, which we always assume to be rep-
:
v resented by the set 0,...,p 1 .
Xi Since the spectacu{lar result−s o}f Bourgain, Glibichuk & Konyagin [8],
Heath-Brown & Konyagin [19] and Konyagin [25] on bounds of expo-
r
a
nential sums
(1) exp(2πiaλ/p), a F∗,
∈ p
λ∈G
X
over small multiplicative subgroups of F∗, there has been a remark-
G p
able progress in this direction, also involving sums over consecutive
powers gi, i = 1,...,N, of elements g F∗, see the survey [17] and
∈ p
Date: February 18, 2014.
2010 Mathematics Subject Classification. 11L40.
Key words and phrases. character sums, intervals, multiplicative subgroups of
finite fields.
1
2 M.-C. CHANGANDIGOR E. SHPARLINSKI
also very recent results of Bourgain [4, 5] and Shkredov [28, 29]. Expo-
nential sums over short segments of consecutive powers g,...,gN of a
fixed element g F∗, have also been studied, see [24, 26] and references
∈ p
therein. However the multiplicative analogues of the sums (1), that is,
the sums
χ(a+λ), a F∗,
∈ p
λ∈G
X
with a nonprincipal multiplicative character χ of F have been resisting
p
all attempts to improve the classical bound
(2) χ(a+λ) √p.
≤
(cid:12) (cid:12)
(cid:12)Xλ∈G (cid:12)
(cid:12) (cid:12)
Note that (2) is instant fr(cid:12)om the Weil(cid:12)bound, see [20, Theorem 11.23],
(cid:12) (cid:12)
if one notices that
T
χ(a+λ) = χ(a+µ(p−1)/T),
p 1
λ∈G − µ∈F∗
X Xp
where T = # (but can also be obtained via elementary arguments).
G
We now recall that Bourgain [3, Section 4] has shown that double
sums over short intervals and short segments of consecutive powers
H N
exp(2πiaxgn/p), 1 a < p 1,
≤ −
x=1 n=1
XX
can be estimated for much smaller values of N than for single sums
over consecutive powers. Here we show that similar mixing can also
be applied to the sums of multiplicative characters and thus lead to
nontrivial estimates of the sums
S (a, , ) = χ(x+aλ), 1 a < p 1,
χ
I G ≤ −
x∈I λ∈G
XX
where = 1,...,H is an interval of H consecutive integers and
F∗I is a{multiplic}ative subgroup of order T for the values of H
G ⊆ p
and T to which previous bounds do not apply. More precisely, one
can immediately estimate the sums S (a, , ) nontrivially if for some
χ
I G
fixed ε > 0 we have H p1/4+ε, by using the Burgess bound, see [20,
≥
Theorem 12.6], or T p1/2+ε, by using (2).
≥
1.2. Main results. Here we obtain a nontrivial estimate in a wider
range of parameters H and T.
DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 3
Theorem 1. For every fixed real ε > 0 there are some δ > 0 and η > 0
such that if H > pε and T > p1/2−δ then for the interval = 1,...,H
and the multiplicative subgroup F∗ of order T, we hIave{ }
G ⊆ p
S (a, , ) = O(HTp−η)
χ
I G
uniformly over a F∗ and nonprincipal multiplicative characters χ of
F . ∈ p
p
We also obtain a similar estimate at the other end of region of H
and T, namely for a very small T and H that is still below the reach
of the Burgess bound (see [20, Theorem 12.6]). In fact in this case we
are able to estimate a more general sums
S (f, , ) = χ(x+f(λ)),
χ
I G
x∈I λ∈G
XX
with a non-constant polynomial f F [X].
p
∈
Theorem 2. For every fixed real ε > 0 and integer d 1 there are
≥
some δ > 0 and η > 0 such that if T > pε and H > p1/4−δ then for the
interval = 1,...,H , the multiplicative subgroup F∗ of order
I { } G ⊆ p
T, we have
S (f, , ) = O(HTp−η)
χ
I G
uniformly over polynomials f F [X] of degree d and nonprincipal
p
multiplicative characters χ of F∈.
p
We also give an explicit version of Theorem 1 in the case when H =
p1/4+o(1) and T = p1/2+o(1), that is, when other methods just start to
fail.
Theorem 3. Let H = p1/4+o(1) and T = p1/2+o(1). Then for the interval
= 1,...,H and the multiplicative subgroup F∗ of order T, we
I { } G ⊆ p
have
S (a, , ) HTp−5/48+o(1)
χ
| I G | ≤
uniformly over a F∗ and nonprincipal multiplicative characters χ of
F . ∈ p
p
Furthermore, we also consider double sums
T (a, ) = χ(a+λ+µ), 1 a < p 1,
χ
G ≤ −
λ,µ∈G
X
where both variables run over a multiplicative subgroup F∗.
G ⊆ p
Using recent estimates of Shkredov [28] on the so-called additive
energy of multiplicative subgroups we also estimate them below the
obvious range T p1/2, where T = # , given by the estimate
≥ G
T (a, ) Tp1/2,
χ
| G | ≤
4 M.-C. CHANGANDIGOR E. SHPARLINSKI
which follows from (2).
Theorem 4. Let T p2/3. Then for the multiplicative subgroup
F∗ of order T, we ha≤ve G ⊆
p
T19/26p1/2+o(1), if T p1/2,
≤
T (a, ) T9/13p27/52+o(1), if p1/2 < T p29/48,
| χ G | ≤  ≤
Tp1/3+o(1), if p29/48 < T p2/3,

≤
uniformly over a F∗ and nonprincipal multiplicative characters χ of
F . ∈ p
p
Note that Theorem 4 nontrivial provided that T p13/33+ε for some
≥
fixed ε > 0.
We also give an application of Theorem 4 to primitive roots modulo
p with few non-zero binary digits. More precisely, let u denote the
p
smallest u such that there exists a primitive root modulo p with u
p
non-zero binary digits. It is shows in [16, Theorem 5] that u 2
p
≤
for all but o(Q/logQ) primes p Q, as Q (note that in [16]
≤ → ∞
the result is formulated only for quadratic non-residues but it is easy
to see that the argument also holds for primitive roots). Instead of
o(Q/logQ), can obtain a slightly more explicit but still rather weak
bound on the size of the exceptional set. Here we show that Theorem 4
implies a rather strong bound on the set of primes p Q for which
≤
u 3 does not hold.
p
≤
Theorem 5. For all but at most Q26/33+o(1) primes p Q, we have
≤
u 3 as Q .
p
≤ → ∞
We also note that one may attempt to treat the sums S (a, , )
χ
I G
and T (a, ) within the general theory of double sums of multiplicative
χ
G
characters, see [6, 7, 11, 12, 15, 21, 22, 23] and references therein.
However it seems that none of the presently known results implies a
nontrivial estimate in the range of Theorems 1 and 4.
2. Preparations
2.1. Notation and general conventions. Throughout the paper, p
always denotes a sufficiently large prime number and χ denotes an
non-principal multiplicative character modulo p. We assume that F is
p
represented by the set 0,...,p 1 .
Furthermore, alwa{ys denot−es a} multiplicative subgroup of F∗ of
G p
order # = T and always denotes the set = 1,...,H .
We alGso assume tIhat f F [X] is a of degIree {d 1. In}particular,
p
∈ ≥
f is not a constant.
DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 5
The notations U = O(V) and U V are both equivalent to the
≪
inequality U cV with some constant c > 0 that may depend on the
| | ≤
real parameter ε > 0 and the integer parameters d 1 and ν 1 and
≥ ≥
is absolute otherwise.
In particular, all our estimates are uniform with respect to the poly-
nomial f and the character χ.
2.2. Bounds of some exponential and character sums. First we
recall the classical result of Davenport and Erd˝os [13], which follows
from the Weil bound of multiplicative character sums, see [20, Theo-
rem 11.23].
Lemma 6. For a fixed integer ν 1 and an integer R < p, we have
≥
2ν
R
χ(v +r) R2νp1/2 +Rνp.
≪
(cid:12) (cid:12)
vX∈Fp(cid:12)Xr=1 (cid:12)
(cid:12) (cid:12)
The following resu(cid:12)lt is a versio(cid:12)n of Lemma 6 with ν = 1 which is
(cid:12) (cid:12)
slightly more precise in this case.
Lemma 7. For any set F and complex numbers α of such that
p v
V ⊆
α 1 for v , we have
v
| | ≤ ∈ V
2
χ(u+v) # p.
(cid:12) (cid:12) ≪ V
uX∈Fp(cid:12)Xv∈V (cid:12)
(cid:12) (cid:12)
Proof. Denotingbyχthec(cid:12)onjugatecha(cid:12)racterandrecallingthatχ(w) =
(cid:12) (cid:12)
χ(w−1) for w F∗, we obtain
∈ p
2
χ(u+v) = α α χ(u+v)χ(u+w).
v w
(cid:12) (cid:12)
uX∈Fp(cid:12)Xv∈V (cid:12) vX,w∈V uX∈Fp
(cid:12) (cid:12)
If v = w the(cid:12)inner sum is(cid:12)equal to p 1. So the total contribution from
(cid:12) (cid:12) −
such terms is O(Mp). Otherwise, we derive
χ(u+v)χ(u+w) = χ(u+v w)χ(u)
−
uX∈Fp uX∈Fp
= χ(u+v w)χ(u) = χ 1+(v w)u−1
− −
u∈F∗ u∈F∗
Xp Xp (cid:0) (cid:1)
= χ(1+u) = χ(1+u) χ(1) = χ(1).
− −
uX∈F∗p uX∈Fp
So the total contribution from such terms is O(M2) = O(Mp) and the
result follows.
⊔⊓
6 M.-C. CHANGANDIGOR E. SHPARLINSKI
We also need the following bound of Bourgain [2, Theorem 1].
Lemma 8. For every fixed real ε > 0 and integer r 1 there is some
≥
ξ > 0 such that for any integers k ,...,k 1 with
1 r
≥
gcd(k ,p 1) < p1−ε, and gcd(k k ,p 1) < p1−ε,
i i j
− − −
for i,j = 1,...,r, i = j, uniformly over the coefficients a ,...,a F ,
1 r p
6 ∈
not all equal to zero, we have
p−1
2πi
exp a xk1 +...+a xkr p1−ξ.
1 r
p ≪
x=1 (cid:18) (cid:19)
X (cid:0) (cid:1)
Clearly for any F F [X] and a multiplicative subgroup F∗ of
∈ p G ⊆ p
order # = T we have
G
p−1
1 2πi 1 2πi
exp F(λ) = exp F(x(p−1)/T) p−ξ.
# p p 1 p ≪
G λ∈G (cid:18) (cid:19) − x=1 (cid:18) (cid:19)
X X
so we derive from Lemma 8:
Corollary 9. For every fixed real ε > 0 and integer d 1 there is
some ξ > 0 such that for T pε, uniformly over a F∗, w≥e have
≥ ∈ p
2πi
exp af(λ) Tp−ξ.
p ≪
λ∈G (cid:18) (cid:19)
X
2.3. Bound on the number of solutions to some congruences.
FirstwenotethatcombiningCorollary9withtheErd˝os-Tur´an inequal-
ity (see, for example, [14, Theorem 1.21]) that relates the uniformity
of distribution to exponential sums, we immediately obtain:
Lemma 10. For every fixed real ε > 0 and integer r 1 there is some
≥
κ > 0 such that for T pε, we have
≥
HT
# λ : f(λ) b+x (mod p), where x = +O T1−κ ,
{ ∈ G ≡ ∈ I} p
uniformly over b F . (cid:0) (cid:1)
p
∈
Let N( , ) be the number of solutions to the congruence
I G
λx y (mod p), x,y , λ .
≡ ∈ I ∈ G
Some of our results rely on an upper bound on N( , ) which is
I G
given in [9, Theorem 1], see also [10] for some other bounds.
Lemma 11. Let ν 1 be a fixed integer. Then
≥
N( , ) Ht(2ν+1)/2ν(ν+1)p−1/2(ν+1)+o(1) +H2t1/νp−1/ν+o(1),
I G ≤
DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 7
as p , where
→ ∞
t = max T,p1/2 .
{ }
We also use the following bound which is due to Ayyad, Cochrane
and Zheng [1, Theorem 1].
Lemma 12. Let = b +1,...,b +h for some integers p > h +b >
i i i i i i
J { }
b 1, i = 1,2,3,4. Then
i
≥
# (x ,x ,x ,x ) : x x x x (mod p)
1 2 3 4 1 2 3 4 1 2 3 4
{ ∈ J ×J ×J ×J ≡ }
1
= h h h h +O (h h h h )1/2(logp)2 .
1 2 3 4 1 2 3 4
p
(cid:16) (cid:17)
We now fix some real L > 1 and denote by the set of primes of
L
the interval [L,2L]. We need an upper bound on the quantity
W = # (u ,u ,ℓ ,ℓ ,s ,s ) 2 2 2 :
1 2 1 2 1 2
∈ I ×L ×S
(3)
n u +s u +s
1 1 2 2
(mod p)
ℓ ≡ ℓ
1 2
o
for some special class of sets.
We say that a set F is h-spaced if no elements s ,s and
p 1 2
S ⊆ ∈ S
positive integer k h satisfy the equality s +k = s .
1 2
≤
The following result is given in [11] and is based on some ideas of
Shao [27].
Lemma 13. If L < H and 2HL < p then for any H-spaced set for
S
W, given by (3) we have
(# HL)2
W S +# HLpo(1).
≪ p S
We also define
(4) U = U(v)2,
vX∈Fp
where
u+f(λ)
(5) U(v) = # (u,ℓ,λ) : v (mod p) .
∈ I ×L×G ℓ ≡
(cid:26) (cid:27)
Lemma 14. For every fixed real ε > 0 and integer d 1 there are
≥
some δ > and η > 0 such that if
T > pε and p1/2−ε H L
≥ ≥
then for U, given by (4) we have
U HLT2p−η.
≪
8 M.-C. CHANGANDIGOR E. SHPARLINSKI
Proof. Let be the largest H-separated subset of = f(λ) : λ
1 0
S F { ∈
. By Lemma 10 we have # pκ for some fixed κ > 0.
1
G} S ≫
Inductively, we define as the largest H-separated subset of
k+1
S
k
= , k = 1,2,....
k k−1 j
F F \ S
j=1
[
Clearly for some b F and a set = b+1,...,b+H we have
p
∈ J { }
#
k
#( ) F .
k
F ∩J ≥ #
k+1
F
On the other hand, by Lemma 10
#( ) #( ) Tp−κ.
k 1
F ∩J ≤ F ∩J ≪
Hence there is a partition
K
=
0 k
F S
k=0
[
into disjoined sets with K Tp−κ/2 such that
≤
# Tp−κ/2,
0
• S ≤
is H-separated with # pκ/2, k = 1,...,K.
k k
• S S ≥
For k = 0,...,K we define
u+s
U (v) = # (u,ℓ,s) : v (mod p) .
k k
∈ I ×L×S ℓ ≡
(cid:26) (cid:27)
We have
K K
U(v) = U (v) = U (v)+ U (v).
k 0 k
k=0 k=1
X X
So, squaring out and summing over all v F , we obtain
p
∈
2
K
U U (v)2 + U (v)
0 k
≪
!
vX∈Fp vX∈Fp Xk=1
K
= U (v)2 + U (v)U (v).
0 k m
vX∈Fp vX∈FpkX,m=1
Now, changing the order of summation in the second term in the above
and then using the Cauchy inequality, yields
(6) U V +V2,
≪ 1 2
DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 9
where
1/2
K
V = U (v)2 and V = U (v)2 .
1 0 2 k
 
vX∈Fp Xk=1 vX∈Fp
 
We have,
V = # (u ,u ,ℓ ,ℓ ,s ,s ) 2 2 2 :
1 1 2 1 2 1 2 ∈ I ×L ×S0
n u +s u +s
1 1 2 2
(mod p)
ℓ ≡ ℓ
1 2
o
max # (u ,u ,ℓ ,ℓ ) 2 2 :
1 2 1 2
≤ s1,s2∈Fp ∈ I ×L
n
u +s u +s
1 1 2 2
(mod p) .
ℓ ≡ ℓ
1 2
o
Since L H p1/2−ε, by Lemma 12 we obtain
≤ ≤
(7) V (# )2HL(logp)2 HLT2p−ε(logp)2.
1 0
≪ S ≪
Furthermore, Lemma 13 implies that for k = 1,...,K we have
U (v)2 (# HL)2p−1 +# HLpo(1).
k k k
≪ S S
vX∈Fp
Hence, applying the Cauchy inequality, we derive
K
V # HLp−1/2 +(# )1/2H1/2L1/2po(1)
2 k k
≪ S S
k=1
X(cid:0) (cid:1)
K
HLTp−1/2 +H1/2L1/2po(1) (# )1/2
k
≤ S
k=1
X
1/2
K
HLTp−1/2 +H1/2L1/2po(1) K #
k
≤ S
!
k=1
X
HLTp−1/2 +H1/2K1/2L1/2T1/2po(1).
≤
Since K Tp−κ/2 and L H p1/3, we see that
≤ ≤ ≤
(8) V HLTp−1/2 +H1/2L1/2Tp−κ/2+o(1) H1/2L1/2Tp−κ/2+o(1)
2
≪ ≤
(assuming that κ is small enough). Substituting (7) and (8) in (6),
leads us to the bound
U HLT2p−εlogp+HLT2p−κ+o(1)
≪
and the result follows.
⊔⊓
10 M.-C. CHANGANDIGOR E. SHPARLINSKI
Let E( ) betheadditive energy ofamultiplicative subgroup F∗,
G G ⊆ p
that is
E( ) = # (λ ,µ ,λ ,µ ) 4 : λ +µ = λ +µ .
1 1 2 2 1 1 2 2
G { ∈ G }
By a result of Heath-Brown and Konyagin [19], if # = T p2/3 we
G ≤
have
E( ) T5/2.
G ≪
Recently, Shkredov [28] has given an improvement which we present
in the following slightly less precise form (which supreses logarithmic
factors in po(1)).
Lemma 15. For T p2/3 we have
≤
T32/13po(1), if T p1/2,
≤
E( ) T31/13p1/26+o(1), if p1/2 < T p29/48,
G ≤  ≤
T3p−1/3+o(1), if p29/48 < T p2/3.

≤
.

3. Proofs of main results
3.1. Proof of Theorem 1. We have
1
(9) S (a, , ) = W,
χ
I G T
where
W = χ(µ)χ(µx+aλ).
x∈I λ,µ∈G
X X
(since χ(µ) = χ(µ−1) for µ F∗). Hence
∈ p
W χ(xµ+aλ) .
| | ≤ (cid:12) (cid:12)
Xx∈I λX,µ∈G(cid:12)Xλ∈G (cid:12)
(cid:12) (cid:12)
Collecting the products µx with th(cid:12)e same value u(cid:12) F , we obtain
(cid:12) (cid:12)∈ p
W R(u) χ(u+aλ) ,
| | ≤ (cid:12) (cid:12)
uX∈Fp (cid:12)Xλ∈G (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
R(u) = # (x,µ) : µx = u .
{ ∈ I ×G }
So, by the Cauchy inequality,
2
W 2 R(u)2 χ(u+aλ) .
| | ≤ (cid:12) (cid:12)
uX∈Fp uX∈Fp(cid:12)Xλ∈G (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)