We shall work on the vector space Fl q, and throughout the paper, we shall assume that the characteristic of the finite fieldFq is sufficiently large so that some minor technical problems ca
Trang 1ON FOUR-VARIABLE EXPANDERS IN FINITE FIELDS∗
LE ANH VINH†
Abstract Let f : F l q → Fq be a given function We say that f is a moderate expander if there exists an > 0 such that for |A| q 1− one has that |f(A, , A)| q We show that
x1x2+ (x3− x4 )2, f (x1) + g(x2) + x3x4, f (x1) + g(x2) + (x3 − x4 )2, f (x1)g(x2) + x3x4 , and
f(x1)g(x2) + (x3− x4 ) 2are moderate expanders with the threshold = 3/8 for various polynomials
f, g ∈ Fq [x].
Key words expanders, distance problems, sum-product estimates AMS subject classification 11B75
DOI 10.1137/120892015
vector space over the fieldF Given a function f : F l → F, define
f (E) = {f(x) : x ∈ E},
the image of f under the subset E We say that f is an l-variable expander with ex-pansion index if |f(E)| ≥ C |E| 1/l+ for every subset E, possibly under some general density or structural assumptions on E The question of whether certain
polynomi-als have the expander property has been studied in various classicial problems For
example, given a finite subset E ⊂ R l, the Erd˝os distance problem [7] deals with the function Δ : Rl × R l → R, where Δ(x, y) = x − y It is conjectured that Δ is a
2l-variable expander with expansion index 1/2l, i.e., |Δ(E, E)| |E| 2/l Here and throughout, X Y means that X ≥ CY for some large constant C This problem in
the Euclidean plane has recently been solved by Guth and Katz [17] They showed
that a set of N points in R2 has at least cN/ log N distinct distances For the latest developments on the Erd˝os distance problem in higher dimensions, see [29, 37] and the references contained therein
Let Fq denote a finite field with q elements, where q, a power of an odd prime,
is viewed as an asymptotic parameter We shall work on the vector space Fl
q, and throughout the paper, we shall assume that the characteristic of the finite fieldFq is sufficiently large so that some minor technical problems can be overcome ForE ⊂ F l
q
(l ≥ 2), the finite analogue of the classical Erd˝os distance problem is to determine the
smallest possible cardinality of the set
Δ(E) = {x − y = (x1− y1)2+· · · + (x l − y l)2:x, y ∈ E} ⊂ F q
The first nontrivial result on the Erd˝os distance problem in vector spaces over finite
fields is due to Bourgain, Katz, and Tao [4], who showed that if q is a prime, q ≡ 3
(mod 4), then for every ε > 0 and E ⊂ F2
q with |E| ≤ C ε q2, there exists δ > 0 such that|Δ(E)| ≥ C δ |E|1+δ for some constants C
ε , C δ The relationship between ε and δ
in their arguments, however, is difficult to determine In addition, it is quite subtle
∗Received by the editors September 20, 2012; accepted for publication (in revised form)
Septem-ber 3, 2013; published electronically DecemSeptem-ber 5, 2013 This research was supported by Vietnam National Foundation for Science and Technology Development grant 102.01-2012.29.
http://www.siam.org/journals/sidma/27-4/89201.html
†University of Education, Vietnam National University, Hanoi, Vietnam (vinhla@vnu.edu.vn).
2038
Trang 2to go up to higher dimensional cases with these arguments Iosevich and Rudnev [27]
used Fourier analytic methods to show that there are absolute constants c1, c2 > 0
such that for any odd prime power q and any set E ⊂ F l
q of cardinality |E| ≥ c1q l/2,
we have
q, q l−12 |E|.
In [43], Vu gave another proof of (1.1) using the graph theoretic method (see also [40] for a similar proof) Iosevich and Rudnev reformulated the question in analogy with the Falconer distance problem: How large does E ⊂ F l
q , l ≥ 2, need to be to
ensure that Δ(E) contains a positive proportion of the elements of F q? The above result implies that if |E| ≥ 2q l+12 then Δ(E) = F q directly in line with Falconer’s result in the Euclidean setting that for a setE with Hausdorff dimension greater than
(l + 1)/2, the distance set is of positive measure At first, it seems reasonable that the exponent (l + 1)/2 may be unprovable, in line with the Falconer distance conjecture
described above However, Hart et al discovered in [25] that the arithmetic of the
problem makes the exponent (l + 1)/2 the best possible in odd dimensions, at least
in general fields In even dimensions, it is still possible that the correct exponent is
l/2, in analogy with the Euclidean case In [5], Chapman et al took a first step in
this direction by showing that ifE ⊂ F2
q satisfies |E| ≥ q 4/3 then |Δ(E)| ≥ cq This
is in line with Wolff’s result [44] for the Falconer conjecture [10] in the plane which says that the Lebesgue measure of the set of distances determined by a subset of the
plane of Hausdorff dimension greater than 4/3 is positive.
Another well-known problem, the sum-product estimate problem, also can be interpreted as a result about expanders This problem deals with the fact that for a given set if one function is nonexpanding then it may imply that another function is
an expander For any subset A, B ⊂ Z, define
A + B = {a + b : a ∈ A, b ∈ B}, A · B = {ab : a ∈ A, b ∈ B}.
Erd˝os [8] conjectured that for any subset A ⊂ Z, either |A+ A| or |A· A| is large, that
is,
max(|A + A|, |A · A|) |A| 2−o(1) .
To support this conjecture, Erd˝os and Szemer´edi [9] only gave the bound
max(|A + A|, |A · A|) |A| 1+
for a positive constant > 0 Explicit bounds on have been studied by many researchers (see [6] and the references therein) The current best known bound ≥
3/14 − δ where δ → 0 as |A| → ∞ is due to Solymosi [34] The sum-product problems
have also been explored in the context of the finite fieldsFq In this context, one may not rely on the topological structure of the real spaces The first nontrivial result is
due to Bourgain, Katz and Tao [4] If A ⊂ F p , p is a prime, and if |A| ≤ p 1− for some > 0, then there exists δ > 0 such that
max(|A + A|, |A · A|) |A| 1+δ . The relationship between and δ in their arguments, however, is difficult to determine.
Quantitative versions of this estimate have been developed by various researchers; see for example [11, 12, 23, 26, 35, 39, 43] and the references therein
Trang 3A related question that has recently received attention is the following Let
A ⊂ F q ; how large does A need to be to make sure that
F∗
q ⊂ dA2= A · A + · · · + A · A (d times)?
When q is a prime, Bourgain [2] showed that if A ⊂ F q of cardinality |A| ≥ Cq 3/4
then
A · A + A · A + A · A = F q
This result also comes from S´ark¨ozy [30] and then is extended by Gyarmati and S´ark¨ozy [18, 19] to arbitrarily finite fields Fq , where q is a large odd prime power.
Glibichuk [15] proved in the case of prime fieldsFp that for d = 8, one can take |A| >
√ q and extended [14] this result to arbitrary finite fields under a weaker assumption.
Furthermore, Schoen and Shkredov [32] reduced Glibichuk’s d = 0 to d = 6 for large
p Glibichuk and Konyagin [16] also proved that if A is a subset ofF∗
q (p prime), and
|A| > p δ , δ > 0, then kA2=Zp with k 41/δ.
In [24, 25], a geometric approach to this problem has been developed In partic-ular, it was proved that if|A| > q 1/2+1/2d thenF∗
q ⊂ dA2, and if|A| > q 1/2+1/2(2d−1)
then |dA2| ≥ q/2 In the most studied case, d = 2, |2A2| q whenever |A| q 2/3 and 2A2 = F∗
q whenever |A| q 3/4 S´ark¨ozy [30, 31] also studied the expanding
property of the set|2A2| and |A + A + A · A| Using additive character sum estimates,
he proved that |A + A + A · A| q whenever |A| q 2/3 and A + A + A · A = F q
whenever|A| q 3/4 These results imply that f1(x1, x2, x3, x4) = x1x2+ x3x4 and
f2(x1, x2, x3, x4) = x1+ x2+ x3x4are four-variable expanders for|A| q 2/3 Garaev
[12] and Garaev and Garcia [13] also considered these expanders over some special sets to obtain new results on the sum-product problem in finite fields The author reproved these results using graph theory methods in [39] The interested reader is referred to [3, 20, 22, 33, 38] for more general results on the subject
1.1 Four-variable expanders In general, expanders can be classified into
three types depending on their expansion rates over the field
Definition 1.1 (see [26, Definition 1.1]) Let f :Fd
q → F q be a given function.
• We say that f is a strong expander with the threshold if for all |A| q 1−
one has that |f(A, , A)| ≥ q − k for a fixed constant k.
• We say that f is a moderate expander with the threshold if for all |A| q 1−
one has that |f(A, , A)| q.
• We say that f is a weak expander with the threshold (, δ) if for all |A| q 1−
one has that |f(A, , A)| |A| δ q 1−δ . The aforementioned results imply that x1+ x2+ x3x4, x1x2+ x3x4, and (x1−
x2)2+ (x3− x4)2 are moderate expanders with the threshold = 1/3 To the best knowledge of the author, there is no known moderate four-variable expander with
the threshold greater than 1/3 In this paper, using graph theoretic methods, we construct various moderate four-variable expanders with the threshold 3/8 More
precisely, we have the following theorem
Theorem 1.2 Let p be the characteristic of Fq , f, g ∈ F q [x], and A ⊆ F q of cardinality |A| q 1/2 .
(1) We have
|{x1x2+ (x3− x4)2: x
1, x2, x3, x4∈ A}| min
q, |A|4
q 3/2
.
Trang 4(2) If 1 ≤ deg(f) < deg(g) < p then
|{f(x1) + g(x2) + x3x4: x1, x2, x3, x4∈ A}| min
q, |A|4
q 3/2
, and
|{f(x1) + g(x2) + (x3− x4)2: x
1, x2, x3, x4∈ A}| min
q, |A|4
q 3/2
.
(3) Suppose f contains some irreducible factors that are not factors of g such that
the great common divisor of the powers of these factors in the canonical factorization
of f is 1, and vice versa Suppose deg(f ) + deg(g) < p Then
|{f(x1)g(x2) + x3x4: x1, x2, x3, x4∈ A}| min
q, |A|4
q 3/2
, and
|{f(x1)g(x2) + (x3− x4)2: x
1, x2, x3, x4∈ A}| min
q, |A|4
q 3/2
.
Theorem 1.2 implies that x1x2+ (x3− x4)2, f (x
1) + g(x2) + x3x4, f (x1) + g(x2) +
(x3− x4)2, f (x
1)g(x2) + x3x4, and f (x1)g(x2) + (x3− x4)2 are moderate expanders
with the threshold 3/8 for various polynomials f, g ∈ F q [x].
q consists of column vectorsx, with jth entry x j ∈ F q Define the distance betweenx, y ∈ F l
q by
x − y =l
j=1
(x j − y j)2.
Given a subsetE ⊂ F l
q, we define the distance set ofE as
Δ(E) = {x − y : x, y ∈ E}.
Similarly, we definte the product set ofE as
Π(E) = {x · y : x, y ∈ E},
wherex · y = x1y1+· · · + x l y lis the usual dot product
Using Fourier analytic methods over finite fields, Iosevich and Rudnev [27] showed that if|E| ≥ 2q (l+1)/2 then Δ(E) ≡ F q Hart and Iosevich [24] improved the threshold
to q l2/(l−1) under the additional assumptions that E has product structure and the
distance set covers a positive proportion of the fieldFq
Theorem 1.3 (see [24, Theorem 1.1, Corollary 1.2]) Suppose that E = E1×
· · · × E l , where E1, , E l ⊂ F q Suppose that
|E| q 2l−1 l2 Then |Δ(E)| q.
Hart et al [25] also obtained a similar result for the product sets in vector spaces over finite fields
Trang 5Theorem 1.4 (see [25, Theorem 2.5]) Suppose that E = E1× · · · × E l , where
E1, , E l ⊂ F q Suppose that
|E| q 2l−1 l2 Then |Π(E)| q.
Theorems 1.3 and 1.4 imply that if A ⊂ F q of cardinality|A| q 2l−1 l then
|Δ(A l)|, |Π(A l)| q.
In this paper, we show that either Δ(A l ) or Π(A l) contain a positive proportion of the field Fq under a weaker condition |A| q 2l+1 4l Note that going from one set formulation in the following theorem to a multiset formulation is just a matter of inserting a different letter in a couple of places
Theorem 1.5 Let A ⊂ F q of cardinality |A| q 1/2 Then, we have
max{|Δ(A l)|, |Π(A l)|} min
q, |A| 2l
q (2l−1)/2
.
In particular, Theorem 1.5 implies that if A ⊂ F q of cardinality|A| q 2l+1 4l then
max{|Δ(A l)|, |Π(A l)|} q.
be the eigenvalues of its adjacency matrix The quantity λ(G) = max {λ2, |λ n |} is
called the second eigenvalue of G A graph G = (V, E) is called an (n, d, λ)-graph
if it is d-regular, has n vertices, and the second eigenvalue of G is at most λ It is well known (see [1, Chapter 9] for more details) that if λ is much smaller than the degree d, then G has certain randomlike properties For two (not necessarily) disjoint subsets of vertices U, W ⊂ V , let e(U, W ) be the number of ordered pairs (u, w) such
that u ∈ U, w ∈ W , and (u, w) is an edge of G We will need the following well-known
fact
Lemma 2.1 (see [1, Corollary 9.2.5]) Let G = (V, E) be an (n, d, λ)-graph For
any two sets U, W ⊂ V , we have
e(U,W) − d |U||W |
n
≤ λ|U||W |.
finite fieldFq is defined as follows The vertex set of the sum-square graphFS q is the set Fq × F q Two vertices a = (a1, a2) and b = (b1, b2)∈ V (FS q) are connected by
an edge, (a, b) ∈ E(FS q ), if and only if a1+ b1 = (a2+ b2)2 We have the following pseudorandomness of the sum-square graphFS q
Lemma 2.2 The graph FS q is an
q2, q,
2q − graph
Proof It is clear that FS q is a regular graph of order q2 and of valency q We now estimate the eigenvalues of this multigraph (i.e., graph with loops) For any
Trang 6a = (a1, a2) = b = (b1, b2) ∈ V (FS q), we count the number of solutions of the following system
a1+ x1= (a2+ x2)2, b1+ x1= (b2+ x2)2, x = (x1, x2)∈ V (FS q ).
The system has a unique solution
x1=
a1− b1
a2− b2 + (a2− b2)
2
/4 − a1,
x2=
a1− b1
a2− b2 − (a2+ b2)
/2
if a2 = b2, and no solution otherwise In other words, two different vertices a =
(a1, a2) and b = (b1, b2) have a unique common vertex if a2 = b2 and no common
vertex otherwise Let M be the adjacency matrix of FS q It follows that
where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of
the graphS E , where V ( S E) =Fq × F q and for any two distinct verticesa, b ∈ V (S E), (a, b) is an edge of S E if and only if a2= b2 It follows thatS E is a (q − 1)-regular
graph Since FS q is a q-regular graph, q is an eigenvalue of M with the all-one
eigenvector 1 The graphFS q is connected, therefore the eigenvalue q has multiplicity
one It is clear that the graphFS q contains (many) triangles which implies that the
graph is not bipartite Hence, for any other eigenvalue θ of the graph FS q, |θ| < q.
Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1 ⊥ , so J v θ = 0
It follows from (2.1) that (θ2 − q + 1)v θ = −Ev θ Since S E is a (q − 1)-regular
graph, absolute values of eigenvalues ofS E are bounded by q − 1 This implies that
θ2≤ 2(q − 1) The lemma follows.
sum-product graphFP q (λ) is defined as follows The vertex set of the sum-product graph
FP q (λ) is the set Fq × F q Two vertices a = (a1, a2) and b = (b1, b2)∈ V (FP q (λ))
are connected by an edge, (a, b) ∈ E(FP q (λ)), if and only if a1+ b1+ a2b2= λ Our
construction is similar to that of Solymosi in [35] We have the following pseudoran-domness of the sum-product graphFP q (λ).
Lemma 2.3 The graph FP q (λ) is an
q2, q,
2q − graph Proof It is clear that FP q (λ) is a regular graph of order q2 and of valency q.
We now estimate the eigenvalues of this multigraph (i.e graph with loops) For any
a = (a1, a2) = b = (b1, b2)∈ V (FP q (λ)), we count the number of solutions of the
following system
a1+ x1+ a2x2= b1+ x1+ b2x2= λ, x = (x1, x2)∈ V (FP q (λ)).
The system has a unique solution
x1= λ − a2b1− a1b2
a2− b2 ,
x2= b1− a1
a2− b2
Trang 7if a2 = b2, and no solution otherwise In other words, two different vertices a =
(a1, a2) and b = (b1, b2) have a unique common vertex if a2 = b2 and no common
vertex otherwise Let M be the adjacency matrix of FP q (λ) It follows that
M2= J + (q − 1)I − E,
where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of
the graphS E , where V ( S E) =Fq × F q and for any two distinct verticesa, b ∈ V (S E), (a, b) is an edge of S E if and only if a2= b2 It follows thatS E is a (q − 1)-regular
graph Since the graph FP q (λ) is a q-regular graph, q is an eigenvalue of M with
the all-one eigenvector 1 The graph FP q (λ) is connected, therefore, the eigenvalue
q has multiplicity one Similar to the proof of Theorem 2.2, for any other eigenvalue
θ of FP q , θ2< 2q − 1 The lemma follows.
3 Three-variable expanders.
result on the three-variable expander x1+ (x2− x3)2.
Theorem 3.1 For any X1, X2, X3⊆ F q , we have
x1+ (x2− x3)2: x
1∈ X1, x2∈ X2, x3∈ X3 min
q, |X1||X2||X3|
q
Proof Let X4 = | x1+ (x2− x3)2: x
1∈ X1, x2∈ X2, x3∈ X3 ⊂ F q Let N
be the number of solutions of equation −x4+ x1+ (x2− x3)2 = 0, (x
1, x2, x3, x4)∈
X1× X2× X3× X4 It is clear that N = |X1||X2||X3| Besides, N is the number
of edges between X4× X2 and (−X1)× (−X3) of the sum-square graph FS q From Lemma 2.1 and Lemma 2.2, we have
|X1||X2||X3| − |X1||X2||X3||X4|
q
≤2q |X1||X2||X3||X4|,
implies
|X1||X2||X3| ≤ |X1||X2||X3||X4|
2q |X1||X2||X3||X4|.
Let t =
|X4| ≥ 0, then
|X1||X2||X3|
2+
2qt −|X1||X2||X3| ≥ 0,
which implies that
|X4| ≥ −
√
2q +
2q + 4 |X1||X2||X3|/q
2
|X1||X2||X3|/q
|X1||X2||X3|
√
2q +
2q + 4 |X1||X2||X3|/q
min
√ q,
|X1||X2||X3| q
.
This concludes the proof of the theorem
Trang 83.2 The expander x + yz From Lemma 2.3, we have the following result on
the three-variable expander x1+ x2x3 The proof of Theorem 3.2 is similar to the proof of Theorem 3.1 and is left for the reader
Theorem 3.2 For any X1, X2, X3⊆ F q , we have
|{x1+ x2x3: x1∈ X1, x2∈ X2, x3∈ X3}| min
q, |X1||X2||X3|
q
.
4 Proof of Theorem 1.2 We first recall a sum-product estimate of Garaev
[12] (see also [35, 39] for different proofs of this result)
Lemma 4.1 (see [12]) For arbitrary sets A, B, C ⊂ F q , we have
|A · B||A + C| min
q |A|, |A|2|B||C|
q
.
Using Fourier analysis techniques, Hart, Li, and Shen [26] obtained the following generalized sum-product-type estimates
Lemma 4.2 (see [26, Theorem 2.6]) Let p be the characteristic of Fq and
f, g ∈ F q [x] For any subsets A, B, C ⊂ F q we have the following.
(1) If 1 ≤ deg(f) < deg(g) < p then
|f(A) + B||g(A) + C| min(|A|q, |A|2|B||C|q −1 ).
Particularly, one has
|f(A) + g(A)| min(|A| 1/2 q 1/2 , |A|2q −1/2 ).
(2) Suppose f contains some irreducible factors that are not factors of g such that
the great common divisor of the powers of these factors in the canonical factorization
of f is 1, and vice versa Suppose deg(f ) + deg(g) < p Then
|f(A) · B||g(A) · C| min(|A|q, |A|2|B||C|q −1 ).
Particularly, one has
|f(A) · g(A)| min(|A| 1/2 q 1/2 , |A|2q −1/2 ).
We are now ready to give a proof of Theorem 1.2 Let
T = {x1x2+ (x3− x4)2: x
1, x2, x3, x4∈ A},
X1={aa : a, a ∈ A},
X
1={(a − a )2: a, a ∈ A},
X2= X3 = A.
Then|X
1| ≥ |A − A|/2 From Lemma 4.1, we have
1| |A · A||A − A| min
q |A|, |A|4 q
.
It follows from Theorem 3.1 that
(4.2)
|T | = |{x1+ (x2− x3)2: x
1∈ X1, x2∈ X2, x3∈ X3}| min
q, |X1||X2||X3|
q
.
Trang 9It follows from Theorem 3.2 that
(4.3) |T | = |{x1+ x2x3: x 1∈ X
1, x2∈ X2, x3∈ X3}| min
q, |X
1||X2||X3| q
.
Putting (4.1), (4.2), and (4.3) together, we have
|T | min
q, |A|4
q 3/2
.
This proves the first part of Theorem 1.2 Similarly, parts (2) and (3) of Theorem 1.2 follow from parts (1) and (2) of Lemma 4.2
q l/(2l−1) then the theorem follows from Theorem 1.3 and Theorem 1.4 Therefore, we can assume that
q 2l+1 4l |A| q 2l−1 l
For the base case l = 2, then q 5/8 |A| q 2/3 Let X = {(a − b)2 : a, b ∈ A},
Y = Z = A It follows from Theorem 3.1 that
(5.1) |Δ(A2)| = x + (y − z)2: x ∈ X, y ∈ Y, z ∈ Z min
q, |X||Y ||Z| q
.
Let X ={ab : a, b ∈ A}, Y = Z = A It follows from Theorem 3.2 that
(5.2) |Π(A2)| = |{x + yz : x ∈ X , y ∈ Y, z ∈ Z}| min
q, |X ||Y ||Z|
q
.
From (4.1), we have
(5.3) |X1||X
1| |A · A||A − A| min
q |A|, |A|4 q
|A|4
q .
Putting (5.1), (5.2), and (5.3) together, we have
max{|Δ(A2)|, |Π(A2)|} min
q, |A|4
q 3/2
.
Suppose the statement holds for l ≥ 2 We show that it also holds for l + 1 By the
induction hypothesis, we have
(5.4) max{|Δ(A l)|, |Π(A l)|} min
q, |A| 2l
q (2l−1)/2
.
From (5.4), Theorems 3.1, and 3.2, we have
max{|Δ(A l+1)|, |Π(A l+1)|} min
q, max
|Δ(A l)||Y ||Z|
|Π(A l)||Y ||Z|
q
min
q, |A|2
q max{|Δ(A l)|, |Π(A l)|}
min
q, |A| 2l+2
q (2l+1)/2
,
concluding the proof of the theorem
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