Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields

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by  Simon Macourt

MULTIPLICATIVE ENERGY OF SHIFTED SUBGROUPS AND
BOUNDS ON EXPONENTIAL SUMS WITH TRINOMIALS IN
FINITE FIELDS
7
1
0
SIMON MACOURT, ILYA D. SHKREDOV, AND IGOR E. SHPARLINSKI
2

n
a Abstract. Wegiveanew boundoncolineartriplesinsubgroups
J
of prime finite fields and use it to give some new bounds on expo-

2
nential sums with trinomials.
2

]
T
N 1. Introduction
.
h 1.1. Set up. For a prime p, we use F to denote the finite field of p
p
t
a elements.
m
For a t-sparse polynomial
[
t

1 Ψ(X) = a Xki
v i
2 Xi=1
9 with some pairwise distinct non-zero integers exponents k ,...,k and
1 1 t
coefficients a ,...,a F∗, and a multiplicative character χ of F∗ we
6 1 t ∈ p p
0 define the sums
1. S (Ψ) = χ(x)e (Ψ(x))
χ p
0
7 xX∈F∗p
1 where e (u) = exp(2πiu/p) and χ is an arbitrary multiplicative char-
p
v: acter of F∗. The most interesting well-studied special case is when
p
i
X χ = χ is a principal character, however most of our results extend
0
r to the general case with any loss of strength of complication of the
a
argument (as well as most of the previous results), so this is how we
present them.
The main challenge here is to estimate these sums better than by
the Weil bound
S (Ψ) max k ,...,k p1/2,
χ 1 t
| | ≤ { }
see [36, Appendix 5, Example 12], by taking advantage of sparsity and
also the arithmetic structure of the exponents k ,...,k . For monomi-
1 t
als Ψ(X) = aXk (where we can always assume that k p 1) the first
| −
2010 Mathematics Subject Classification. 11L07, 11T23.
Key words and phrases. exponential sum, sparse polynomial, trinomial.
1
2 S. MACOURT,I. D.SHKREDOV,ANDI. E. SHPARLINSKI
bound of this type is due to Shparlinski [34] which has then been im-
proved and extended in various directions by Bourgain, Glibichuk and
Konyagin [7], Bourgain [3], Heath-Brown and Konyagin [19], Konya-
gin [22], Shkredov [27], Shteinikov [32].
Akulinichev [1] gives several bounds on binomials, see also [38].
Cochrane, Coffelt and Pinner, see [9, 10, 11, 12, 13, 14] and references
therein, have given a series of other bounds on exponential sums with
sparse polynomials, some of which we present below in Section 1.2.
We also remark that exponential sums with sparse polynomials and
a composite denominator have been studied in [4, 33].
Here we use a slightly different approach to improve some of the
previous results. Our approach is related on reducing bounds of expo-
nential sumswithsparsepolynomialstoboundsofweighted multilinear
exponential sums of the type considered in [25]. However, instead of
applying the results of [25] directly, we first obtain their more precise
version for triple weighted sums over multiplicative subgroups of F∗,
p
which could be of independent interest, see Lemma 3.4 below.
This result rests on an extension of the bound on the number of
collinear triples from multiplicative subgroups from [28, Proposition 1]
to subgroups of any size, see Theorem 1.1. In turn, this gives a new
bound on the multiplicative energy of arbitrary subgroups, see Corol-
lary 4.1, and has several other applications, see Section 4.
Although here we concentrate on the case of trinomials
(1.1) Ψ(X) = aXk +bXm +cXn,
our method works, without any changes, for more general sums with
polynomials of the shape
Ψ(X) = aXk +f(Xm)+g(Xn)
with arbitrary polynomials f,g F [X] (uniformly in the degrees of f
p
∈
and g, which essentially means that they can be any functions defined
F
on ).
p
One can certainly use our approach for sums with quadrinomials,
using a version of [25, Theorem 1.3], using our Lemma 3.2 in an appro-
priate place. Furthermore, using results of [4, 6, 17], one can consider
the case of arbitrary sparse polynomials.
The notation A B is equivalent to A c B for some constant
≪ | | ≤ | |
c, which, throughout the paper may only depend on the number of
monomials in the sparse polynomials under considerations.
EXPONENTIAL SUMS WITH TRINOMIALS 3
1.2. Previousresults. Wecompareourresults fortrinomials(1.1)with
the estimates of Cochrane, Coffelt and Pinner [9, Equation (1.6)]
1/4
kℓm
(1.2) S (Ψ) p7/8
χ
≪ (cid:18)max k,ℓ,m (cid:19)
{ }
which is non-trivial for min kℓ,km,ℓm < p1/2, and of Cochrane and
{ }
Pinner [11, Theorem 1.1]:
(1.3) S (Ψ) p5/6(kℓm)1/9
χ
≪
which is non-trivial for kℓm < p3/2, or ℓm < p1/2.
We also recall the bound of Cochrane, Coffelt and Pinner [10, Corol-
lary 1.1]
(1.4) S (Ψ) D1/2p7/8 +(kℓm)1/4p5/8
χ
≪
where D = gcd(k,ℓ,m,p 1).
−
1.3. New results. For a set F∗ we define T( ) to be number of
U ⊂ p U
solutions to
u v u v
1 1 2 2
(1.5) − = − , u ,v ,w , i = 1,2.
i i i
u w u w ∈ U
1 1 2 2
− −
As we relation (2.1) shows, the triples (u ,v ,w ), i = 1,2 satisfy-
i i i
ing (1.5) defined there collinear points. Recent results on the quantity
T( ) for an arbitrary set can be found in [23].
U U
Note that in (1.5) as well as in all similar expressions of this type
we consider only the values of the variables for which these expressions
are defined (that is, u = w , i = 1,2 in (1.5)).
i i
6
We beginby providing a new result onthenumber of collinear triples
in subgroups.
Theorem 1.1. Let be a multiplicative subgroup of F∗. Then
G p
p 3, if p3/4,
6
|G| |G| ≥
T( ) |G|  5p−1/2, if p3/4 > p1/2logp,
G − p ≪  |G|4log , if < p|1G/2|l≥ogp.
|G| |G| |G|

Remark 1.2. Theorem 1.1 is new only for subgroups of size > p1/2,
|G|
otherwise it is contained in [28, Proposition 1].
Remark1.3. Analysing the proof of Theorem 1.1, one see the Cartesian
products can replaced with the Cartesian products of different
G × G
cosets of and also to variables u ,v ,u λ , i = 1,2, from two
i i i i
G ∈ G
different cosets of .
G
We obtain the following result on trinomial sums.
4 S. MACOURT,I. D.SHKREDOV,ANDI. E. SHPARLINSKI
Theorem 1.4. Let Ψ(X) be a trinomial of the form (1.1) with a F∗.
∈ p
Define
d = gcd(k,p 1), e = gcd(ℓ,p 1), f = gcd(m,p 1)
− − −
and
d e
g = , h = .
gcd(d,f) gcd(e,f)
Suppose g h, then
≥
p7/8f1/8, if h p1/2logp,
≥
S (Ψ)  p15/16(f/h)1/8+o(1), if g p1/2logp > h,
χ
≪  p(f/gh)1/8+o(1), if g ≥< p1/2logp.
We observe that the bound of Theorem 1.4 is independent of the
size of our powers k, ℓ and m and is strongest for large d and e
and small greatest common divisors f, gcd(d,f) and gcd(e,f). In
particular, it may remain nontrivial even for polynomials of very large
degrees, while the bounds (1.2), (1.3) and (1.4) all become trivial for
trinomials of large degree. Thus it is easy to give various families of
set of parameters when Theorem 1.4 improves the bounds (1.2), (1.3)
and (1.4) simultaneously.
2. Colinear Triples
2.1. Prelimaries. We require some previous results. The first one is a
result of Mit’kin [24, Theorem 2] extending that of Heath-Brown and
Konyagin [19, Lemma 5], see also [22, 26, 31] for further generalisations
generalisations.
Lemma 2.1. Let and be subgroups of F∗ and let = F∗/ and
= F∗/ be Gthe setsHof distinct coset reppresentatiMvesG of p aGnd
Min HF∗. FoprHan arbitrary set Θ such that G H
p ⊂ MG ×MH
2 2 Θ < p3 and Θ ,
|G| |H| | | | | ≤ |G||H|
we have
(x,y) 2 : ux+vy = 1 ( Θ 2)1/3.
{ ∈ G } ≪ |G||H|| |
(uX,v)∈Θ(cid:12) (cid:12)
(cid:12) (cid:12)
We note that we use Lemma 2.1 only for = , however we present
G H
it in full generality as we believe the results has be known better.
Given a line
ℓ = (x,y) F2 : y = ax+b
a,b { ∈ p }
for some non-zero pair (a,b) F2 and a set F , we denote
∈ p A ⊆ p
ι (ℓ ) = ℓ ( ) .
A a,b a,b
| ∩ A×A |
EXPONENTIAL SUMS WITH TRINOMIALS 5
The following elementary identities are well-known and no doubt
have appeared, implicitly and explicitly, in a number of works.
Lemma 2.2. Let F . Then
p
A ⊆
ι (ℓ ) = p 2 and ι (ℓ )2 = 4 3 +p 2.
A a,b A a,b
|A| |A| −|A| |A|
(aX,b)∈F2 (aX,b)∈F2
p p
Proof. The first relationis obvious as forevery (x,y,a) 2 F there
p
∈ A ×
is a unique b = y ax counted in that sum.
−
For the second sum, we write
ι (ℓ )2
A a,b
(aX,b)∈F2
p
= # (a,b) F2 : v = au+b, y = ax+b .
{ ∈ p }
(u,v,Xx,y)∈A4
We now note that 2 quadruples (u,v,x,y) 4 with (u,v) = (x,y)
define exactly p p|aAir|s (a,b) = (a,v au) ∈FA F . Furthermore,
p p
− ∈ ×
2( 1) quadruples (u,v,x,y) 4 with u = x but v = y do
|A| |A|− ∈ A 6
not define any pairs (a,b). The remaining
4 2( 1) 2 = 4 3
|A| −|A| |A|− −|A| |A| −|A|
pairs define one pair (a,b) F2 each, which concludes the proof.
∈ p ⊔⊓
We now immediately derive
Corollary 2.3. Let F . Then
p
A ⊆
2 2
ι (ℓ ) |A| p 2.
A a,b
(cid:18) − p (cid:19) ≤ |A|
(aX,b)∈F2
p
We now link T( ) with the quantities ι (ℓ ).
G a,b
G
Lemma 2.4. Let F . Then
p
A ⊆
T( ) = ι (ℓ )3 +O 4 .
A a,b
A |A|
(aX,b)∈F2 (cid:0) (cid:1)
p
Proof. Transforming the equation (1.5) into
u v u w
1 1 1 1
− = − u ,v ,w , i = 1,2,
i i i
u v u w ∈ U
2 2 2 2
− −
we introduce an error of magnitude O( 4) (coming from different
|A|
pairs of variables which must be distinct). Then collecting, for every
a F , the solutions with
p
∈
u v u w
1 1 1 1
− = − = a
u v u w
2 2 2 2
− −
6 S. MACOURT,I. D.SHKREDOV,ANDI. E. SHPARLINSKI
we derive:
u au = v av = w aw .
1 2 1 2 1 2
− − −
We now denoting this common value by b and observe that for any
(a,b) F2 there are ι (ℓ )3 solutions to
∈ p A a,b
(2.1) u au = v av = w aw = b.
1 2 1 2 1 2
− − −
Summing over all pairs (a,b) F2, we obtain the result.
∈ p ⊔⊓
Using the identity X3 = X(X Y)2 + 2X2Y XY2 with X =
− −
ι (ℓ ) and Y = 2/p we see that
A a,b
|A|
2 2
ι (ℓ )3 = ι (ℓ ) ι (ℓ ) |A|
A a,b A a,b A a,b
(cid:18) − p (cid:19)
(aX,b)∈F2 (aX,b)∈F2
p p
2 4
+2|A| ι (ℓ )2 |A| ι (ℓ ).
p A a,b − p2 A a,b
(aX,b)∈F2 (aX,b)∈F2
p p
Combining this with Lemma 2.2 yields
ι (ℓ )3
A a,b
(aX,b)∈F2
p
6 2 2
= |A| + ι (ℓ ) ι (ℓ ) |A| 2 5/p+2 4.
A a,b A a,b
p (cid:18) − p (cid:19) − |A| |A|
(aX,b)∈F2
p
Hence using Lemma 2.4, we obtain
Corollary 2.5. Let F . Then
p
A ⊆
6 2 2
T( ) |A| = ι (ℓ ) ι (ℓ ) |A| +O( 4).
A a,b A a,b
A − p (cid:18) − p (cid:19) |A|
(aX,b)∈F2
p
Given two set , F , we define E×( , ) to bethe multiplicative
p
U V ⊆ U V
energy of and , that is, the number of solutions to
U V
u v = u v , u ,u , v ,v .
1 1 2 2 1 2 1 2
∈ U ∈ V
For = we also write
U V
E×( ) = E×( , ).
U U U
It is easy to see that for a subgroup of F∗ we have
G ⊆ p
(2.2) T( ) = 2E×( 1) = E×( g, h).
G |G| G − G − G −
X
g,h∈G
Finally, we need the following bound which is a special case of [28,
Proposition 1].
EXPONENTIAL SUMS WITH TRINOMIALS 7
Lemma 2.6. Let p be a prime number and let be a subgroup of F∗
G p
with < p1/2. Then
|G|
T( ) 4log .
G ≪ |G| |G|
2.2. ProofofTheorem 1.1. Nowwearereadytoproveourmainresult.
The arguments follows [28, 30].
First of all, notethat Lemma 2.6 implies the required result provided
< √p.
|G|Let be the set of all multiplicative characters of F∗ and let
X p
∗ = χ be the set of all nonprincipal characters of F∗, see [21,
X X \{ 0} p
Chapter 3] for a background on characters.
Using the orthogonality of characters (and collecting together solu-
tions with at one, thus at least two variables which are equal to 1) we
write
1 (u 1)(u 1)
E×( 1) = χ 1 − 2 − +O( 2)
G − p 1 (cid:18)(u 1)(u 1)(cid:19) |G|
u1,u2,uX3,u4∈G\{1} − χX∈X 3 − 4 −
4
1
= (cid:12) χ(u 1)(cid:12) +O 2
p 1 (cid:12) − (cid:12) |G|
− χX∈X (cid:12)u∈XG\{1} (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
(cid:12) (cid:12) 4
4 1(cid:12) (cid:12)
= |G| + χ(u 1) +O( 2)
(cid:12) (cid:12)
p p 1 − |G|
− χX∈X∗(cid:12)(cid:12)Xu∈G (cid:12)(cid:12)
(cid:12) (cid:12)
(we note that the condition u =(cid:12)1 can be drop(cid:12)ped from the summation
6
also a slight simplification in main term can be absorbed in the error
term O( 2)). Now for k = (p 1)/ we have
|G| − |G|
1
(2.3) χ(u 1) = (cid:12) χ xk 1 (cid:12) p1/2
(cid:12) (cid:12)
− k (cid:12) − (cid:12) ≤
(cid:12)(cid:12)Xu∈G (cid:12)(cid:12) (cid:12)(cid:12)xX∈F∗p (cid:0) (cid:1)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
by the Weil b(cid:12)ound on mul(cid:12)tiplica(cid:12)tive character (cid:12)sums, see [21, Theo-
rem 11.23]. Hence, using this bound and then the of characters one
more time, we obtain
4 2
1 p
χ(u 1) χ(u 1) p2 .
(cid:12) (cid:12) (cid:12) (cid:12)
p 1 − ≪ p 1 − ≪ |G|
− χX∈X∗(cid:12)(cid:12)Xu∈G (cid:12)(cid:12) − χX∈X∗(cid:12)(cid:12)Xu∈G (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
whichdomina(cid:12)testhepreviou(cid:12)serrorterm O((cid:12) 2) andgive(cid:12)s E×( 1) =
|G| G−
4/p+O(p ). Thus, together with (2.2)
|G| |G|
4 6
(2.4) T( ) = 2 |G| +O(p ) = |G| +O p 3
G |G| (cid:18) p |G| (cid:19) p |G|
(cid:0) (cid:1)
8 S. MACOURT,I. D.SHKREDOV,ANDI. E. SHPARLINSKI
This gives the desired result for > p3/4.
|G|
Let ∆ 3 be a parameter to be chosen later. Using Corollary 2.5,
≥
we obtain
6
(2.5) T( ) |G| 4 +∆ 2p+Σ,
G − p ≪ |G| |G|
where
2 2
Σ = ι (ℓ ) ι (ℓ ) |G| .
A a,b G a,b
(cid:18) − p (cid:19)
(aX,b)∈F2
p
ιG(ℓa,b)>∆
Clearly, the contribution to Σ from lines with ab = 0, is most 4 as
|G|
in this case it does not vanish only for either a or b and then
∈ G ∈ G
we have ι (ℓ ) = and
A a,b
|G|
2 2
ι (ℓ ) ι (ℓ ) |G| = O 4 .
A a,b G a,b
(cid:18) − p (cid:19) |G|
(aX,b)∈F2 (cid:0) (cid:1)
p
ab=0
Thus
(2.6) Σ = Σ∗ +O 4
|G|
(cid:0) (cid:1)
where
2 2
Σ∗ = ι (ℓ ) ι (ℓ ) |G| .
A a,b G a,b
(cid:18) − p (cid:19)
(a,bX)∈(F∗)2
p
ιG(ℓa,b)>∆
Hence, we can assume below that i(l) ∆ and have deal just with
≥
the remaining sum Σ.
Returning to (1.5), we see that the quantity T( ) equals the number
G
of the solution to the equation
(2.7) (u v )(u w ) = (u w )(u v ), u ,v ,w , i = 1,2.
1 1 2 2 1 1 2 2 i i i
− − − − ∈ G
One can assume that the products in (2.7) are nonzero and u = v be-
1 1
6
cause there are at most O( 4) such solutions, which can be absorbed
|G|
in the error term in (2.5). Denote by σ the remaining number of the
solutions.
Let, as before, = F∗/ be set of distinct coset representatives
of in F∗. Take aMnoGther pparGameter τ ∆ and put
G p ≥
Θ = (α,β) 2 : (x,y) 2 : αx+βy = 1 τ .
τ { ∈ MG |{ ∈ G }| ≥ }
In other words, Θ is the set of (u,v) 2 for which the lines
τ ∈ MG
(2.8) Lα,β = {(x,y) ∈ F2p : αx+βy = 1} = ℓ−αβ−1,β−1
EXPONENTIAL SUMS WITH TRINOMIALS 9
have the intersection with 2 at least
G
ιG (ℓ−αβ−1,β−1) τ.
≥
By Lemma 2.1, we have Θ τ ( Θ )2/3 provided
τ τ
| | ≪ |G|| |
4 Θ < p3 and Θ 2.
τ τ
|G| | | | | ≤ |G|
Clearly, this immediately implies that for the set
= (α,β) F∗ 2 : (x,y) 2 : αx+βy = 1 τ
Qτ { ∈ p |{ ∈ G }| ≥ }
(cid:0) (cid:1)
we have
(2.9) = 2 Θ 4τ−3
τ τ
|Q | |G| | | ≪ |G|
provided 4 Θ < p3 and Θ 2.
τ τ
|G| | | | | ≤ |G|
Since 2 > p, we have
|G|
Θ 2 = (p 1)2/ −2 −2
τ G
| | ≤ |M | − |G| ≤ |G|
we see that the upper bound (2.9) for holds provided the only
τ
Q
condition 4 Θ < p3 satisfied. Suppose, in contrary, that Θ >
τ τ
|G| | | | |
p3/ 4. Whence, the number of incidences between points of = 2
|G| P G
and the lines as above with (α,β) is at least
α,β τ
L ∈ Q
(2.10) τ > p3 −2∆.
τ
|Q | |G|
On the other hand, by a classical result which holds over any field (see,
for example [8, Corollary 5.2] or [37, Exercise 8.2.1]) the number of
incidences for any set of points and a set of lines is at most
τ
P Q
1/2 + . Hence
τ τ
|Q | |P| |Q |
(2.11) τ 1/2 +
τ τ τ
|Q | ≤ |Q | |P| |Q |
and we obtain
(2.12) τ2 2 = 4.
τ
|Q | ≪ |P| |G|
Combining (2.10) and (2.12), we derive
(2.13) p3 −2∆ < τ 4τ−1 4∆−1.
τ
|G| |Q | ≪ |G| ≤ |G|
Now taking
(2.14) ∆ = c 3p−3/2,
|G|
for a sufficiently large constant c (and recalling that p1/2), we
|G| ≥
see that (2.13) impossible.
Thus, for the above choice (2.14) of ∆ we have the desired condition
4 Θ < p3 for any τ ∆ and therefore the bound (2.9) holds,
τ
|G| | | ≥
10 S. MACOURT,I. D.SHKREDOV,ANDI. E. SHPARLINSKI
We see from (2.8) that there is a one-to-one correspondence between
the lines ℓ , (a,b) F∗ 2 and the lines , (α,β) F∗ 2. We
a,b ∈ p Lα,β ∈ p
now now define (cid:0) (cid:1) (cid:0) (cid:1)
τ = ej∆, j = 0,1,...,J.
j
Note that due to the choice of ∆ and the condition p1/2 we have
|G| ≥
τ τ = ∆ 3p−3/2, j = 0,1,...,J.
j 0
≥ ≫ |G|
Then, recalling the bound (2.9), we conclude that the contribution to
Σ from the lines with τ ι (ℓ ) > τ is bounded by
∗ j+1 G a,b j
≥
(2.15) τ τ + 2/p 2 τ3 . 4
Qτj j+1 j+1 |G| ≪ Qτj j+1 ≪ |G|
(cid:12) (cid:12) (cid:0) (cid:1) (cid:12) (cid:12)
Summing up (2(cid:12).15)(cid:12)we obtain (cid:12) (cid:12)
Σ∗ J 4 4log .
≪ |G| ≪ |G| |G|
Substituting this bound in (2.6) and combining it with (2.5), we obtain
6
T( ) = |G| +O 5p−1/2 + 4log
G p |G| |G| |G|
(cid:0) (cid:1)
in the range p3/4 > p1/2. Thus, together with (2.4) (for >
|G| ≥ |G|
p3/4) and Lemma 2.6 ( (for < p1/2) we conclude the proof.
|G|
Remark 2.7. In principle, a stronger version of the classical incidence
bound which is used (2.11) may lead to improvements of Theorem 1.1.
However the range where such improvements are known are far away
from the range which appears in our applications, see [35].
3. Trinomial sums
3.1. Preliminaries. We recall the following classical bound of bilinear
sums, see, for example, [5, Equation 1.4] or [17, Lemma 4.1].
Lemma 3.1. For any sets , F and any α = (α ) , β =
p x x∈X
X Y ⊆
(β ) , with
y y∈Y
α 2 = A and β 2 = B,
x y
| | | |
X X
x∈X y∈Y
we have
α β e (xy) pAB.
(cid:12) x y p (cid:12)
≤
(cid:12)(cid:12)xX∈X Xy∈Y (cid:12)(cid:12) p
(cid:12) (cid:12)
We define D ( )(cid:12) to be the number o(cid:12)f solutions of
×
U
(u v )(u v ) = (u v )(u v ), u ,v , i = 1,2,3,4,
1 1 2 2 3 3 4 4 i i
− − − − ∈ U
and as before we define T( ) as the number of solutions to (1.5).
U
We now recall the following bound from [25, Lemma 2.7].