On the volume set of boxes in vector spaces over finite fields

On the volume set of boxes in vector spaces over finite fields tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án,...

On the Volume Set of Boxes in Vector Spaces

Over Finite Fields

ABSTRACT Let Fq be a finite field of q elements We show

that ifA ⊂ F qof cardinality|A| ≳ q1/2, then

|(A − A) · (A − A) · · (A − A)| ≳min

(

q, | A|2

q1/2n−1

)

,

where the product is takenntimes We also obtain similar

re-sults in the setting of finite cyclic rings

1 INTRODUCTION The classical Erd˝os distance problem asks for the minimal number of distinct distances determined by a finite point set in Rn, n ≥ 2 This problem in the Euclidean plane has recently been solved by Guth and Katz ([6]), who show that

a set of N points in R2 has at leastcN/logN distinct distances (For the latest developments on the Erd˝os distance problem in higher dimensions, see [10,13] and the references contained therein.)

Throughout the paper,q = p r (r ≥1) wherep is a sufficiently large prime andr is a fixed constant so that some minor technical problems can be overcome Here and throughout, the notation X ≲ Y means that there exists a constant

C >0 that is independent ofXandY, such thatX ≤ CY LetFqdenote a finite field with qelements For E ⊂ Fn

q (n ≥ 2), the finite analogue of the classical Erd˝os distance problem is to determine the smallest possible cardinality of the set

∆ Fq (E) = {kx − yk = (x1− y1)2+ · · · + (x n − y n )2:x, y ∈ E} ⊂ F q

The first non-trivial result on the Erd˝os distance problem in vector spaces over finite fields is due to Bourgain, Katz, and Tao ([3]), who showed that if q is a

2125

Indiana University Mathematics Journal c , Vol 65, No 6 (2016)

prime,q ≡3 (mod 4), then for everyε >0 andE ⊂ F 2

qwith|E| ≤ C ε q2−ε, there existsδ >0 such that|∆ Fq (E)| ≥ C δ|E| 1/2+δ for some constantsC ε , C δ Iosevich and Rudnev ([9]) used Fourier analytic methods to show that there are absolute constantsc1, c2>0 such that, for any odd prime powerqand any setE ⊂ Fdof cardinality|E| ≥ c1q n/2, we have

(1.1) |∆ Fq (E)| ≥ c2min{q, q (n−1)/2 |E|}.

Iosevich and Rudnev ([9]) reformulated the question in analogy with the Falconer distance problem: how large doesE ⊂ Fn

q,n ≥2, need to be to ensure that∆Fq (E)

contains a positive proportion of the elements of Fq? The above bound meets with Falconer’s result in the Euclidean setting: that for a set E with Hausdorff dimension greater than (n +1)/2 the distance set is of positive measure (For current results of this problem, see [4,15] and the references contained therein.)

In [5], Covert, Iosevich, and Pakianathan extended (1.1) to the setting of finite cyclic ringsZq = Z/qZ(q = p r,pis a sufficiently large prime) Here, the distance set ofE ⊂ Zdis defined similarly by

∆ Zq (E) = {kx − yk = (x1− y1)2+ · · · + (x n − y n )2:x, y ∈ E} ⊂ Z q

In [5], Covert, Iosevich, and Pakianathan obtained a nearly sharp bound for the distance problem inZd More precisely, they proved that ifE ⊂ Zn

q of cardinality

|E| ≳ r (r +1)q (2 r −1)n/(2r )+1/(2r ),

then

Z×q ⊂ ∆ Zq (E).

In [17], the second listed author extended this result using graph-theoretic meth-ods, and showed, roughly speaking, that any sufficiently large subset E ⊂ Zn

q

determines all possible non-degeneratek-simplices

The main purpose of this paper is to study the analogue of the Erd˝os distance problem for the volume set of boxes in finite vector spaces (see also [11] for the analogue problem over the real numbers) In [8], Hart, Iosevich, and Solymosi obtained the following result

Theorem 1.1 ([8, Theorem 1.4]) Let A ⊆ F q , a finite field with q elements Suppose that |A| ≥ Cq1/2 + 1/(2n)with a sufficiently large absolute constant C Then,

(A − A) · (A − A) · · (A − A) = F q , where the product is taken n times.

DenoteVn (A) = (A − A) · (A − A) · · (A − A), where the product is taken

ntimes Balog ([2]) obtained the following results

Theorem 1.2 ([2, Theorem 1.1]) Let A ⊆ F q with |A| ≥ q1/2 + 1/2k

, where

k >1 We haveV 2k+1(A) = F q

Corollary 1.3 ([2, Corollary 1]) Let A ⊆ F q with |A| > q1/2 We have

|Vk (A)| ≥ q1 − 1/2k

.

Using the graph theoretic method and Corollary1.3, we obtain the following theorem

Theorem 1.4. Let A ⊆ F q of cardinality |A| ≳ q1/2 We have

|Vn (A)| ≳min

(

q, | A|2

q1/2n−1

)

.

Specifically, Theorem1.4implies that ifA ⊆ F qof cardinality|A| ≳ q1/2 + 1/2n

, then|Vn (A)| ≳ q Similarly, we also prove the following result over finite rings

Theorem 1.5. Let A ⊆ Z q such that |A| ≳ q1 − 1/(2r ); then,

|Vn (A)| ≳min

(

p r , |A|2

r p r −1+ 1/2n−1

)

.

Using the graph-theoretic method and Theorem1.4, we obtain the following improvement of Theorem1.2

Theorem 1.6. Let A ⊆ F q with |A| ≳ q1/2 + 1/(3 · 2k−1) , where k >1 We have

V 2k+1(A) = F q

Similarly, we also prove the following result over finite rings

Theorem 1.7. Let A ⊆ Z q , a finite ring with q elements Suppose that

|A| ≳p2r q1 − 1/(2r )+1/(3r ·2k−1);

then,Z×q ⊆ V 2k+1(A)

Note that the study of dot-products has a flavor of sum-product theory For related results of this problem, see [7,14] and the references contained therein

2 SOMELEMMAS

In this section, we follow closely [2, Section 2] to extend several sum-product type results to the setting of cyclic rings LetZq = Z/qZ(q = p r,r is a sufficiently large prime) be a finite cyclic ring of orderq DenoteZ×q the set of units ofZqand

Z0q= Zq\ Z ×

q

Lemma 2.1. Let A, B ⊆ Z q with |A| |B| > q2 − 1/r We have

Z×q ⊆ A − A

(B − B) \ Z0

q

.

Proof For any x ∈ Z×

q, consider the setA − xB ⊆ Z q Since

|A − xB| < p r < | A| |B|

p r −1 ,

there exist p r −1 different pairs (a1, b1), (a2, b2), ., (a p r −1, b p r −1) such that

a i − xb i = a j − xb j, orx(b i − b j ) = a i − a j

Ifa i = a j, thenb i = b j, soa j 6= a jwithi 6= j LetM = {a1, a2, , a p r −1 } Since |M − M| > |M| = p r −1, there are a i , a j such that a i − a j ∈ Z ×

q, or

Now, we recall an important tool in additive combinatorics, namely, Ruzsa’s triangle inequality ([12])

Lemma 2.2. Let U, V , W be finite subsets of an arbitrary group G We have

|UV− 1| |W | ≤ |UW | |VW |.

From Lemmas2.1and2.2we prove the following result over finite rings

Lemma 2.3. Let A ⊆ Z q with |A| ≥ q1 − 1/(2r ) We have|Vn (A)| ≥ q1 − 1/(r2n ) Proof Inserting U = V = (A−A)\Z0

q,W = V n, andG = Z qinto Lemma2.2,

it follows from Lemma 2.1 that if |A| > p r −1/2 , then |UV− 1| ≥ p r − p r −1 Therefore,

(p r − p r −1)|V n | ≤ |V n+1|2,

and by induction we get

(p r − p r −1)2n− 1|V| ≤ |V n+1 | 2n

.

Finally, using the trivial bounds|V| = |(A − A) \ Z0

q | > p r −1/2 − p r −1, we have

|Vn (A)| ≥ |V n | ≥ cq1 − 1/(r2n ) ❐

3 PRODUCT-SUM GRAPHS For a graphG, letλ1≥ λ2≥ · · · ≥ λ nbe the eigenvalues of its adjacency matrix The quantityλ(G) =max{λ2, −λ n}is called the second eigenvalue ofG A graph

G = (V, E)is called an(n, d, λ)-graph if it isd-regular, has nvertices, and the second eigenvalue ofGis at mostλ

It is well known ([1, Chapter 9]) that ifλis much smaller than the degreed, thenGhas certain random-like properties More precisely, we have the following lemma

Lemma 3.1 ([1, Corollary 9.2.5]) Let G = (V, E) be an (n, d, λ) -graph 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 Then,

e(B, C) − d|B| |C|

n

≤ λ

q

|B| |C|.

For anyδ ∈ F×

q, the product-sum graphPSq (δ) is defined as follows The vertex set of the product graph PSq (δ) is the setF∗q × Fq Also, the two ver-tices a = (a1, a2)and b = (b1, b2) ∈ V(PS q (δ)) are connected by an edge,

{a, b} ∈ E(PS q (λ)), if and only ifa1b1(a2+ b2) = δ In [16], the second listed author proved a pseudo-randomness of the product graphPSq (δ) Note that, in the statement of [16, Theorem 3.6], all but the largest eigenvalue of the graph

PSq (δ)are bounded byp3q However, looking carefully at the “error graph”E

in the proof of that theorem, we can bound all but the largest eigenvalue ofE by

2√q As a consequence, we have a slight improvement of [16, Theorem 3.6] as

follows

Theorem 3.2 ([16, Theorem 3.6]) For any δ ∈ F×

q , the graphPSq (δ) is a ((q −1)q, q −1,q

2q )-graph.

In this paper, we will study an extension of the above theorem to the setting

of finite cyclic rings Suppose thatq = p r for a sufficiently large primep For anyδ ∈ Z×

q, the product-sum graphPSRq (δ)is defined as follows The vertex set of the sum-product graphPSRq (δ)is the setV (PSR q (δ)) = Z×q × Zq Two vertices a = (a1, a2) and b = (b1, b2) ∈ V(PSR q (δ)) are connected by an edge inE(PSR q (δ)), if and only ifa1b1(a2+ b2) = δ We have the following pseudo-randomness of the product-sum graphPSRq (δ)

Theorem 3.3. For any δ ∈ Z×q , the product-sum graphPSRq (δ) is a

(p2r − p2r −1, p r − p r −1 ,q

2r p2r −1)-graph. Proof It is easy to see thatPSRq (δ)is a regular graph of orderp2r − p2r −1

and valency p r − p r −1 We now compute the eigenvalues of this multigraph (i.e., graphs with loops) Thus, for any two distinct vertices a = (a1, a2)and

b= (b1, b2) ∈ V(PSR q (δ)), we count the number of paths of length two from

atob More precisely, we count the number of solutions of the system

(3.1) a1x1(a2+ x2) = δ, b1x1(b2+ x2) = δ, (x1, x2) ∈ Z×

p r× Zp r

For each solutionx1∈ Z ×

p r of the equation (3.2) x1(a2− b2) = a δ

1 −b δ 1

,

there exists a uniquex2∈ Zp r satisfying the system of equations (3.1) Therefore,

we only need to count the number of solutions of equation (3.2)

For any primep, the p-adic valuationv p (n)of an nonzero integer nis the highest power ofp that dividesn Let α = v p (a2− b2); then, clearly we have

0 ≤ α ≤ r −1 (otherwise, α = r; then, a2 = b2 anda1 = b1) If we have

v p (δ/a1− δ/b1) 6= α, then it is easy to see that equation 3.2has no solution Suppose thatv p (δ/a1− δ/b1) = α Let

γ = (δ/a1− δ/b1)/p α , β = (a2− b2)/p α∈ Z×p r −α

Then, there exists a unique solutionx1∈ Zp r −α ofβx1= γ Putting this back in

to equation (3.2) gives usp αsolutions Hence, the system of equation (3.1) has

p αsolutions ifv p (δ/a1− δ/b1) = α, and no solution otherwise

Therefore, suppose that we have any two distinct verticesa = (a1, a2)and

b = (b1, b2) ∈ V(PSR q (δ)) Let also α = v p (a2− b2); then, (a1, a2)and

(b1, b2)havep αcommon neighbors ifv p (δ/a1− δ/b1) = α, and no common neighbor otherwise LetAbe the adjacency matrix ofPSRq (δ) It follows that

(3.3) A2= J + (p r − p r −1 −1)I −

r −1X

α=0

E α+

r −1X

α=1

(p α−1)F α ,

where we have the following:

• Jis the all-one matrix

• Iis the identity matrix

• E α is the adjacency matrix of the graph B E,α, where the vertex set

ofB E,α isZ×q × Zq Furthermore, for any two verticesU = (a, b) and

V = (c, d) ∈ V(B E,α ), (U, V ) is an edge of B E,α if and only if α =

v p (a2− b2) 6= v p (δ/a1− δ/b1)

• F α is the adjacency matrix of the graphB F,α, where the vertex set

ofB F,α isZ×q × Zq Furthermore, for any two vertices U = (a, b) and

V = (c, d) ∈ V(B F,α ), (U, V ) is an edge of B F,α if and only if α =

v p (a2− b2) = v p (δ/a1− δ/b1)

For anyα >0,B E,αis a regular graph of order less than

(p r −α − p r −α−1 )(p r − p r −α + p r −α−1 ),

and B F,α is a regular graph of order less than (p r −α − p r −α−1)2 Hence, all eigenvalues of E α are at most (p r −α − p r −α−1 )(p r − p r −α + p r −α−1 ), and all eigenvalues ofF αare at most(p r −α − p r −α−1)2 Note thatE0is a zero matrix SincePSRq (δ)is a (p r − p r −1)-regular graph,p r − p r −1 is an eigenvalue

ofAwith the all-one eigenvector 1 The graphPSRq (δ)is connected; therefore, the eigenvaluep r −p r −1has multiplicity one Since the graphPSRq (δ)contains (many) triangles, it is not bipartite Hence,|θ| < p r − p r −1for any other eigen-valueθ Letvθ denote the corresponding eigenvector ofθ Note thatvθ ∈1⊥,

soJv θ=0 It follows from (3.3) that

(θ2− p r + p r −1+1)v θ=−

r −X 1

α=0

E α+

r −X 1

α=1

(p α−1)F α



v θ

Hence,vθis also an eigenvalue of

−

r −1X

α=0

E α+

r −1X

α=1

(p α−1)F α .

Since eigenvalues of the sum of the matrices are bounded by the sum of the largest eigenvalues of the summands, we have

θ2≤ p r − p r −1−1+

r −X 1

α=0

(p r −α − p r −α−1)(p r − p r −α + p r −α−1)

+

r −X 1

α=1

(p α−1)(p r −α − p r −α−1)2.

This implies thatθ2 ≤2r p2r −1, and the lemma follows ❐

4 PROOF OFTHEOREM 1.4AND THEOREM 1.6

4.1 Proof of Theorem 1.4 As a consequence of Theorem3.2, we have the following lemma

Lemma 4.1. For any A ⊆ F×

q , B, C ⊆ F q with |B| >1, we have

|{a(b − c):a ∈ A, b ∈ B, c ∈ C}| ≳min

(

q, | A| |B| |C|

q

)

.

Proof Let D = {a(b − c): a ∈ A, b ∈ B, c ∈ C} ∩ F×q, and letN be the number of solutions of equationad(b − c) =1,(a, b, c, d) ∈ A × B × C × D− 1

It is clear thatN = |A| |B| |C| − |A| |B ∩ C| Besides,Nis the number of edges between two vertex setsA × BandD− 1× (−C)of the product-sum graphPSq From Lemma3.1and Theorem3.2, we have

|A| |B| |C| − |A| |B ∩ C| − |A| |B| |C| |D| q

≤

q

2q|A| |B| |C| |D|,

or equivalently,

|A| |B| |C| − |A| |B ∩ C| ≤ |A| |B| |C| |D| q +q2q|A| |B| |C| |D|.

Lett =p|D| ≥0; then,

p

|A| |B| |C|

2 +q2q t −q|A| |B| |C| +

s

|A|

|B| |C| |B ∩ C| ≥0,

which implies that

q

|D| ≥ −

p

2q +q2q +4|A| |B| |C|/q −4|A| |B ∩ C|/q

2p|A| |B| |C|/q

≥ −

p

2q +p2q + |A| |B| |C|/q

2p|A| |B| |C|/q

1 p

|A| |B| |C|

p

2q +p2q + |A| |B| |C|/q

≳min

(

pq,s|A| |B| |C|

q

)

,

where the third line follows from the fact that|B| |C| ≥2|C| > 43|B ∩ C| This

We are now ready to give a proof of Theorem 1.4 PuttingA = V n−1 (A),

B = C = Ainto Lemma4.1, and using Corollary1.3, we have

|Vn (A)| ≳min

(

q, | A|2

q1/2n−1

)

,

concluding the proof of the theorem

4.2 Proof of Theorem 1.6 We need the following lemma.

Lemma 4.2. Let A, B ⊆ F∗

q and C, D ⊆ F q , such that |A| |B| |C| |D| ≥2q3.

We have AB(C − D) = F q

Proof Let H δ = {(a, b, c, d) ∈ A × B × C × D : ab(c − d) = δ} for any

δ ∈ F∗

q Then,|H δ|is the number of edges between two vertex setsA × C and

B × Dof the product-sum graphPSq (δ) From Lemma3.1and Theorem3.2, we

|H δ| −|A| |B| |C| |D| q

≤

q

2q|A| |B| |C| |D|,

or equivalently,

|H δ| ≥ |A| |B| |C| |D| q −q2q|A| |B| |C| |D|.

Therefore, if|A| |B| |C| |D| >2q3, then|H δ | >0 This impliesAB(C −D) = F q,

We are now ready to give a proof of Theorem 1.6 Putting A = V k (A),

B = V k (A),C = D = Ainto Lemma4.2, and using Theorem1.4, we have that if

|A| ≥ cq1/2 + 1/(3 · 2k−1), thenV 2k+1(A) = F q, concluding the proof of the theorem

5 PROOF OFTHEOREM 1.5AND THEOREM 1.7

5.1 Proof of Theorem 1.5.

Lemma 5.1. Let q = p r with p an odd prime and r ≥2an integer For any

A, B, C ⊆ Z q such that |B| >4|Z 0

q |/3, we have

|a(b − c):a ∈ A, b ∈ B, c ∈ C| ≳min

(

p r , | A| |B| |C|

2r p2r −1

)

.

Proof Let D = {a(b − c): a ∈ A, b ∈ B, c ∈ C} ∩ Z×

q, and letN be the number of solutions of equationad(b − c) =1,(a, b, c, d) ∈ A × B × C × D− 1

It is clear that

N = |A| |B| |C| − |A| X

t∈Z0

q

|B ∩ (C + t)|.

Moreover,Nis the number of edges between two vertex setsA×BandD− 1×(−C)

of the product-sum graphPSRq (δ) From Lemma3.1and Theorem3.3, we get

|A| |B| |C| − |A|

X

t∈Z0

q

|B ∩ (C + t)| − |A| |B| |C| |D| p r

≤q2r p2r −1|A| |B| |C| |D|,

or equivalently,

p

|A| |B| |C| |D|

p r +q2r p2r −1|D|

−q|A| |B| |C| +

s

|A|

|B| |C|

X

t∈Z0

q

|B ∩ (C + t)| ≥0.

Letx =p|D| ≥0; then,

p

|A| |B| |C|

p r x2+q2r p2r −1x

−q|A| |B| |C| +

s

|A|

|B| |C|

X

t∈Z0

q

|B ∩ (C + t)| ≥0,

which implies that

q

q

2r p2r −1

2p|A| |B| |C| p −r

+

v

u2r p2r −1 +4|A| |B| |C|p −r−4|A| X

t∈Z0

q

|B ∩ (C + t)|p −r

2p|A| |B| |C| p −r

≥ −

q

2r p2r −1 +q2r p2r −1+ |A| |B| |C|p −r

2p|A| |B| |C| p −r

=

p

|A| |B| |C|

2(q

2r p2r −1 +q2r p2r −1+ |A| |B| |C| p −r )

≳min(q

p r ,

s

|A| |B| |C|

2r p2r −1

)

,

where the third line follows from the fact that|B| |C| >4|Z 0

q | |C| ≥4|B ∩ C|

We are now ready to give a proof of Theorem 1.5 Putting A = V k (A),

B = C = Ainto Lemma5.1, and using Lemma2.3, we have

|Vn (A)| ≳min

(

p r , |A|2

2r p r −1+ 1/2n−1

)

,

concluding the proof of the theorem

5.2 Proof of Theorem 1.7 Similarly, we need the following lemma over

finite cyclic rings

Lemma 5.2. Let A, B ⊆ Z×

q and C, D ⊆ Z q , such that |A| |B| |C| |D| ≥

2r q4 − 1/r We haveZ×q ⊆ AB(C − D)

Proof Let H δ = {(a, b, c, d) ∈ A × B × C × D : ab(c − d) = δ} for any

δ ∈ Z×

q Then, |H δ|is the number of edges between two vertex setsA × C and

B × Dof the product-sum graphPSRq (δ) From Lemma3.1and Theorem3.3,

we have 

|H δ| −|A| |B| |C| |D| p r

≤

q

2r p2r −1|A| |B| |C| |D|,

or equivalently,

|H δ| ≥ |A| |B| |C| |D| p r −q2r p2r −1|A| |B| |C| |D|.

,

concluding the proof of the theorem

5.2 Proof of Theorem 1.7 Similarly, we need the following lemma over< /b>

finite cyclic rings

Lemma...

)

,

concluding the proof of the theorem

4.2 Proof of Theorem 1.6 We need the following lemma.

Lemma 4.2. Let... q, concluding the proof of the theorem

5 PROOF OF< /small>THEOREM 1.5AND THEOREM 1.7

5.1 Proof of Theorem 1.5.