We also obtain a similar result in the setting of the finite cyclic ring Z/mZ.. Using this result, Bourgain, Katz and Tao [3] proved a theorem of Szemerédi–Trotter type in two-dimensiona
Trang 1Contents lists available atScienceDirect
Discrete Applied Mathematics
journal homepage:www.elsevier.com/locate/dam
On point-line incidences in vector spaces over finite fields
Le Anh Vinh
University of Education, Vietnam National University, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Received 31 August 2012
Received in revised form 2 May 2014
Accepted 18 May 2014
Available online 16 June 2014
Keywords:
Point-line incidences
Szemerédi–Trotter theorem
Residue rings
a b s t r a c t Let Fqbe the finite field of q elements We show that for almost every point setPand line setLin F2
qof cardinality|P | = |L|&q, there exists a pair(p,l) ∈ P × Lwith p∈l We
also obtain a similar result in the setting of the finite cyclic ring Z/mZ.
© 2014 Elsevier B.V All rights reserved
1 Introduction
Let Fq be a finite field of q elements where q is a large odd prime power LetAbe a non-empty subset of a finite field Fq
We consider the sum set
A+A:= {a+b:a,b∈A}
and the product set
A.A:= {a.b:a,b∈A}
Let|A|denote the cardinality ofA Bourgain, Katz and Tao [3] showed that when 1≪ |A| ≪q then
max(|A+A| , |A.A| ) ≫ |A|;
this improves the easy bound|A+A||A.A| & |A| (Here, and throughout, X . Y means that there exists C > 0 such
that X ≤ CY , and X ≪ Y means that X = o(Y).) Using this result, Bourgain, Katz and Tao [3] proved a theorem of Szemerédi–Trotter type in two-dimensional finite field geometries Roughly speaking, this theorem asserts that if we are in the finite plane F2and one has N lines and N points in that plane for some 1≪N≪q2, then there are at most O(N3 / 2 − ϵ)
incidences; this improves the standard bound of O(N3 / 2)obtained from extremal graph theory
In [11], the author proceeded in an opposite direction: the author first proved a theorem of Szemerédi–Trotter type about the number of incidences between points and lines in finite field geometries; then apply this result to obtain a different proof
of a result of Garaev [5] on a sum–product estimate for large subsets of finite fields This estimate is the best known bound
in the finite field problem More precisely, we have the following results
Theorem 1.1 ([ 11 , Theorem 3]) LetPbe a collection of points andLbe a collection of lines in F2 Then we have
{ (p,l) ∈P×L:p∈l} − |P||L|
q
≤q1/ 2
E-mail addresses:leanhvinh@gmail.com , vinhla@math.harvard.edu
http://dx.doi.org/10.1016/j.dam.2014.05.024
0166-218X/ © 2014 Elsevier B.V All rights reserved.
Trang 2In the spirit of Bourgain–Katz–Tao’s result, one can derive fromTheorem 1.1a reasonably good estimate when d = 2 and 1< α <2 LetP be a collection of points andLbe a collection of lines in F2 Suppose that|P| , |L| ≤N=qαwith
1+ ϵ ≤ α ≤2− ϵfor someϵ >0 Then we have
|{ (p,l) ∈P×L:p∈l}| ≤2N3−ϵ4. (1.2) This upper bound has applications in several combinatorial problems, see for example [6,7,11] One can also ask for the lower bound of incidences between a point setPand a line setL It follows fromTheorem 1.1that
|{ (p,l) ∈P×L:p∈l}| ≥ |P||L|
1 / 2
|P||L|
This implies that{ (p,l) ∈P×L:p∈l} ̸≡ ∅when|P||L|&q3 On the other hand, it is possible to construct about q3/ 2
lines and q3 / 2points in F2without any incidences Take q= p2, with the field identified additively with F2 Note that we need to have(Fp,0)be a subfield of Fqto make sure that the product of two elements in(Fp,0)is in(Fp,0) Let y=ax+b with a in(Fp,0)and b in the set(Fp,B)with B having about p/2 elements So, there are about p3lines For each of these lines,
whenever x is in(Fp,0),y has no values in(Fp,B c) Thus, there is a point set and a line set, each with about q3/ 2elements, and no incidences
In the case of incidences between a random point set and a random line set, we however can improve the bound q3/ 2in the sense that for anyα ∈ (0,1)it suffices to take s≥Cαq randomly chosen lines and points in the plane to guarantee that
the probability of no incidences is exponentially smallαs when q is large enough.
Theorem 1.2 For anyα >0, there exist an integer q0=q(α)and a number Cα>0 with the following property When a point setPand a line setLwhere|P| = |L| =s≥Cαq, are chosen randomly in F2, the probability of{ (p,l) ∈P×L:p∈l} ≡ ∅
is at mostαs , provided that q≥q0.
Let m be a large non-prime integer and Z m be the ring of residues mod m Letγ (m)be the smallest prime divisor of
m, τ(m)be the number of divisors of m, andφ(m)be Euler’s totient function We identify Zmwith{0,1, ,m−1} Define the set of units and the set of nonunits in Zqby Z×mand Z0mrespectively The finite Euclidean space Zd mconsists of column
vectors x, with jth entry x j ∈ Zm In [10], Thang and the author extendedTheorem 1.1to the setting of finite cyclic rings
Zm(see also [4] for some related results) One reason for considering this situation is that if one is interested in answering similar questions on the setting of rational points, one can ask questions for such sets and how they compare to the answers
in Rd By scale invariance of these questions, the problem for a subsetEof Qdwould be the same as for subsets of Zd
m More precisely, Thang and the author proved the following theorem on point-line incidences in vector spaces over finite rings
Theorem 1.3 ([ 10 , Theorem 1.6]) LetPbe a collection of points andLbe a collection of lines in Z2m Then we have
{ (p,l) ∈P×L:p∈l} − |P||L|
m
≤ 2τ(m)m2
φ(m)γ (m)1 / 2
Notice thatTheorem 1.3is most effective when m has only few prime divisors For example, if m =p r, in the spirit of
Bourgain–Katz–Tao’s result, one can obtain a reasonably good estimate when p 2r− 1 + ϵ.N.p 2r− ϵ LetPbe a collection of
points andLbe a collection of lines in Z2
r Suppose that|P| , |L|.N=pαwith 2r−1+ ϵ ≤ α ≤2r− ϵfor someϵ >0 Then we have
|{ (p,l) ∈P×L:p∈l}|.N3−4rϵ. (1.4) For the lower bound,Theorem 1.3implies that{ (p,l) ∈P×L:p∈l} ̸≡ ∅when
|P||L| > (φ(4(τ(m))2m6
m))2γ (m) . Our next result is the finite ring analog ofTheorem 1.2 More precisely, we show that for almost every point setPand line setLof cardinality|P| = |L|&m, there exists a pair(p,l) ∈P×Lwith p∈l.
Theorem 1.4 For anyα >0, there exist an integer m0=m(α)and a number Cα>0 with the following property When a point setPand a line setLwhere|P| = |L| =s≥Cαq, are chosen randomly in Z2
m , the probability of{ (p,l) ∈P×L:p∈l} ≡ ∅
is at mostαs , provided that m≥m andγ (m)&m c for some constant c>0.
Trang 32 Pseudo-random graphs
For a graph G, letλ1 ≥ λ2 ≥ · · · ≥ λnbe the eigenvalues of its adjacency matrix The quantityλ(G) =max{ λ2, −λn}
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 mostλ It is well known (see [2, Chapter 9] for more details) that ifλis much smaller than the
degree d, then G has certain random-like 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 For a vertexvof G, let
N(v)denote the set of vertices of G adjacent tovand let d(v)denote its degree Similarly, for a subset U of the vertex set, let N U(v) =N(v) ∩U and d U(v) = |N U(v)| We first recall the following well-known fact (see, for example, [2])
Lemma 2.1 ([ 2 , Corollary 9.2.5]) Let G= (V,E)be an(n,d, λ)-graph For any two sets B,C⊂V , we have
e(B,C) −d|B||C|
n
≤ λ|B||C|
Let G(X,Y)be a bipartite graph We denote the number of edges going through X and Y by e(X,Y) The average degree
¯
d(G)of G is defined to be e(X,Y)/(|X| + |Y| ) It has been shown that if G = G(X,X)has certain random-like properties
and one chooses a sufficiently large random subset S ⊂ X then the probability of G(S,S)being empty is very low More precisely, we have the following lemma due to Hoi Nguyen [9
Lemma 2.2 ([ 9 , Lemma 8]) Let{G n =G(V n,V n)}∞
n= 1be a sequence of bipartite graphs with|V n| → ∞as n→ ∞ Assume that for anyϵ >0, there exist an integerv(ϵ)and a number c(ϵ) >0 such that
e(A,A) ≥c(ϵ)|A|2d¯ (G n)/|V n|
for all|V n| ≥ v(ϵ)and all A⊂V n satisfying|A| ≥ ϵ|V n| Then for anyα >0, there exist an integerv(α)and a number C(α)
with the following property If one chooses a random subset S of V n of cardinality s then the probability of G(S,S)being empty is
at mostαs provided that s≥C(α)|V n| /¯d(G)and|V n| ≥ v(α).
To study the point-line incidences between a point setPand a line setF, we mimic the proof of [9, Lemma 8] to obtain
a two-set version of the above lemma
Lemma 2.3 Let{G n=G(V n,U n)}∞
n= 1be a sequence of bipartite graphs with|V n| = |U n| → ∞as n→ ∞ Assume that for any
ϵ >0, there exist an integerv(ϵ)and a number c(ϵ) >0 such that
e(A,B) ≥c(ϵ)|A||B|¯d(G n)/|V n|
for all|V n| = |U n| ≥ v(ϵ)and all A⊂V n,B⊂U n satisfying|A||B| ≥ ϵ|V n|2 Then for anyα >0, there exist an integerv(α)
and a number C(α)with the following property If one chooses a random subset S of V n of cardinality s and a random subset T
of U n of the same cardinality s, then the probability of G(S,T)being empty is at mostαs provided that s≥C(α)|V n| /¯d(G)and
|V n| ≥ v(α).
Proof We shall view S= { v1, , vs}and T= {u1, ,u s}as ordered random subsets, whose elements will be chosen in order,v1,u1, v2,u2, , vs,u s
For 1 ≤ k ≤ s− 1, let R k be the set of neighbors of the first k+ 1 chosen vertices from V n , i.e R k = {u ∈
U n | (vi,u) ∈ E(G n)for some i ≤ k + 1}and L k be the set of neighbors of the first k chosen vertices from U n, i.e
L k = { v ∈ V n | (v,u i) ∈ E(G n)for some i ≤ k} If G(S,T)is empty, we havevk+ 1 ̸∈ L k and u k+ 1 ̸∈ R k Let A k+ 1be the set of possible choices forvk+ 1from V n\{ v1, , vk}such that|R k+ 1\R k| ≤ c(ϵ)ϵ¯d(G), whereϵ < 1 will be chosen
small enough later and c(ϵ)is the constant from the lemma Similarly, let B k+ 1be the set of possible choices for u k+ 1from
U n\{u1, ,u k}such that|L k+ 1\L k| ≤c(ϵ)ϵ¯d(G)
We first assume that|A k+ 1||B k+ 1| > ϵ|V n|2for some 1 ≤ k ≤ s−1 Since|A k+ 1| ≤ |V n| , |B k+ 1| > ϵ|V n| Besides
B k+ 1∩R k= ∅, so we have
e(A k+ 1,B k+ 1) ≤e(A k+ 1,U n\R k) ≤c(ϵ)ϵ¯d(G)|A k+ 1| <c(ϵ)|A k+ 1||B k+ 1|¯d(G)/|V n| (2.1)
Here, the second inequality follows from the property of the set A k+ 1that any vertex in A k+ 1has at most c(ϵ)ϵ¯d(G)neighbors
in U n\R k
On the other hand, it follows from the property of G nthat
e(A k+ 1,B k+ 1) ≥c(ϵ)|A k+ 1||B k+ 1|¯d(G n)/|V n| , (2.2)
provided that n is large enough Putting(2.1)and(2.2)together, we have a contradiction This implies that if G(S,T)is empty then|A ||B | ≤ ϵ|V|2for 1≤k≤s−1
Trang 4Let s be sufficiently large, for example, s≥ 4(c(ϵ)ϵ)− 1|V n| /¯d(G n), and assume thatv1,u1, v2,u2, , vs,u shave been
chosen Let svbe the number of verticesvk+ 1that do not belong to A k+ 1and s u be the number of vertices u k+ 1that do not
belong to B k+ 1 We have
|U n| ≥ |R s| ≥
vk+ 1 ̸∈A k+ 1
|R k+ 1\R k| ≥svc(ϵ)ϵ¯d(G n).
Hence, sv≤ (c(ϵ)ϵ)− 1|U n| /¯d(G n) ≤s/4 Similarly, we also have s u≤s/4 This implies that there are s−sv−s u≥s/2 pairs (vk+ 1,u k+ 1)such thatvk+ 1 ∈A k+ 1and u k+ 1 ∈B k+ 1 Since|A k+ 1||B k+ 1| ≤ ϵ|V n|2for 1≤k≤s−1, the number of subsets
S⊂V n,T ⊂U n such that G(S,T)is empty is bounded by
sv,s u≤s/ 4
s
sv
s
s u
|V n|2(s u+sv)(ϵ|V n|2)s−sv−s u ≤
sv,s u≤s/ 4
s
sv
s
s u
|V n|2sϵs/ 2
<4s|V n|2sϵs/ 2< (256ϵ)s/ 2|V n| (|V n| −1) (|V n| −s+1)|U n| (|U n| −1) (|U n| −s+1).
Takingϵ = α2/256, the lemma follows
3 Erdős–Rényi graphs
3.1 Erdős–Rényi graph over a finite field
We recall a well-known construction of Alon and Krivelevich [1] Let PG(q,d) denote the projective geometry of
dimension d−1 over the finite field Fq The vertices of PG(q,d)correspond to the equivalence classes of the set of all
non-zero vectors x= (x1, ,x d)over Fq, where two vectors are equivalent if one is a multiple of the other by an element
of the field LetE R(Fd)denote the graph whose vertices are the points of PG(q,d)and two (not necessarily distinct) vertices
x and y are adjacent if and only if x1y1+ · · · +x d y d =0 This construction is well known In the case d= 2, this graph is called the Erdös–Rényi graph It is easy to see that the number of vertices ofE R(Fd)is n q,d= (q d−1)/(q−1)and that it is
d q,d -regular for d q,d= (q d− 1−1)/(q−1) The eigenvalues ofE R(Fd)are easy to compute [1] Let A be the adjacency matrix
ofE R(Fd) Then, by properties of PG(q,d),A2 =AA T = µJ+ (d q,d− µ)I, whereµ = (q d− 2−1)/(q−1),J is the all one matrix and I is the identity matrix, both of size n q,d×n q,d Thus the largest eigenvalue of A is d q,dand the absolute values of all other eigenvalues ared q,d− µ =q(d− 2 )/ 2 Precisely, we have just proved the following lemma
Lemma 3.1 ([ 1 ]) For any odd prime power q and d≥2, the Erdős–Rényi graphE R(Fd)is an
q d−1
q−1,q d−1−1
q−1 ,q(d− 2 )/ 2
−graph.
3.2 Erdős–Rényi graph over a finite ring
Similarly, we have a variant of Erdős–Rényi graph over the ring Zm We define the projective space PG(m,d)over Zmas
follows For any x∈Zd m\ (Z0m)d, we denote[x]the equivalence class of x in Z d m\ (Z0m)d , where x,y∈Zd m\ (Z0m)dare equivalent
if and only if x = ty for some t ∈ Z×
m LetE R(Zd m)denote the Erdős–Rényi graph whose vertices are the points of the
projective space PG(m,d)over Zm, where two vertices[x]and[y]are connected if and only if x·y=0 In [10], Thang and the author obtained the following pseudo-randomness of the Erdős–Rényi graphE R(Zd m)
Lemma 3.2 ([ 10 , Theorem 2.4]) For any m,d≥2, the Erdős–Rényi graphE R(Zd m)is an
m d− (m− φ(m))d
φ(m) ,
m d− 1− (m− φ(m))d− 1
2τ(m)m d− 1
φ(m)γ (m)(d− 2 )/ 2
−graph.
In order to proveLemma 3.2, Thang and the author first studied the spectrum of the zero-product graph in Zm For any
integers m,d≥2, the zero-product graphZPm,dis defined as follows The vertex set of the zero-product graphZPm,dis
the set V(ZPm,d) =Zd m\ (Z0m)d Two vertices a and b∈V(ZPm,d)are connected by an edge,(a,b) ∈E(ZPm,d), if and only
if a·b=0 Thang and the author proved that for any m,d≥2, the zero-product graphZPm,dis an
m d− (m− φ(m))d,m d−1− (m− φ(m))d− 1, γ (2τ(m)m d−1
m)(d− 2 )/ 2
Notice that the product-graph can be obtained from the Erdős–Rényi graph by blowing it up (which means replacing each vertex by an independent set of sizeφ(m)and connecting two vertices in the new graph if and only if the corresponding vertices of the Erdős–Rényi graph are connected by an edge) The bound inLemma 3.2can be derived immediately from (3.1)by using well known results on the eigenvalues of the tensor product of two matrices (see [8,10] for more details)
Trang 54 Proof of the main results
4.1 Proof of Theorem 1.2
Let PG(q,3)be the projective plane over the finite field Fq LetGq, 3be a bipartite graph with the vertex set PG(q,3) ×
PG(q,3)and the edge set
{ ([x] , [y] ) ∈PG(q,d) ×PG(q,d) :x·y=x1y1+ · · ·x d y d=0}
Note that the graphGq, 3is just a bipartite version of the Erdős–Rényi graph in Section3.1 It follows thatGq, 3is a regular bipartite graph of valencyd¯ (Gq, 3) = (q2−1)/(q−1) Besides, fromLemmas 2.1and3.1, for anyE,F ⊂PG(q,3), we have
e(E,F) −q2−1
q3−1|E||F|
≤q1/ 2
We can embed the plane F2into PF3by identifying(x,y)with the equivalence class of(x,y,1) Any line in F2also can
be represented uniquely as an equivalence class in PF3of some non-zero element h∈F3 For each x∈F3, we denote[x]
the equivalence class of x in PF3 The relation
x·y=x1y1+x2y2+x3y3=0
is equivalent to the fact that the points represented by[x]and[y]lie on the lines represented by[y]and[x], respectively Hence, the number of point-line incidences in F2can be interpreted as the number of edges between two vertex sets ofGq, 3 For anyϵ >0, |E||F| ≥ ϵq4and q≥2/ϵ, it follows from(4.1)that
e(E,F) ≥ q2−1
q3−1|E||F| −q
1 / 2
|E||F| > q2−1
2(q3−1) |E||F| =
¯
d(Gq,d)
2|V(Gq,d)| |E||F| .
Let c(ϵ) = 1/2 andv(ϵ) ≥ 8/ϵ2,Theorem 1.2now follows fromLemma 2.3 Notice that,Theorem 1.1also follows immediately from Eq.(4.1)
4.2 Proof of Theorem 1.4
Let PG(m,d)be the projective plane over the finite ring Zm LetPm,d be a bipartite graph with the vertex set PG(m,d) ×
PG(m,d)and the edge set
{ ([x] , [y] ) ∈PG(m,d) ×PG(m,d) :x·y=x1y1+ · · ·x d y d=0}
Note that the graphPq, 3is just a bipartite version of the Erdős–Rényi graph in Section3.2 It follows thatPq, 3is a regular bipartite graph of valency
¯
d(Pq, 3) =m2− (m− φ(m))2
φ(m) .
We can embed the plane Z2m into PZ3mby identifying(x,y)with the equivalence class of(x,y,1) Any line in Z2malso can
be represented uniquely as an equivalence class in PZ3
m of some non-zero element h∈Z3
m For each x∈Z3
m, we denote[x]
the equivalence class of x in PZ3
m The relation
x·y=x1y1+x2y2+x3y3=0
is equivalent to the fact that the points represented by[x]and[y]lie on the lines represented by[y]and[x], respectively Hence, the number of point-line incidences in Z2
mcan be interpreted as the number of edges between two vertex sets of
Pm, 3
FromLemmas 2.1and3.2, for anyE,F ⊂PG(m,3)we have
e(E,F) −m2− (m− φ(m))2
m3− (m− φ(m))3|E||F|
≤ 2τ(m)m2
φ(m)γ (m)1 / 2
Hence, it is easy to check that, for anyϵ >0, if|E||F|&ϵm4and
(φ(m))2γ (m)
m2(τ(m))2 > 16
then
e(E,F) ≥ d¯ (Pq, 3)
2|V(Pq, 3)| |E||F| .
Let c(ϵ) =1/2, v(ϵ)such that for any m≥ v(ϵ)andγ (m)&m c for some constant c>0 then Eq.(4.3)holds,Theorem 1.4 now follows fromLemma 2.3 Notice that,Theorem 1.3also follows immediately from Eq.(4.2)
Trang 6This research was supported by Vietnam National University- Hanoi project QGTD.13.02
References
[1] N Alon, M Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs Combin 13 (1997) 217–225.
[2] N Alon, J.H Spencer, The Probabilistic Method, second ed., Willey-Interscience, 2000.
[3] J Bourgain, N Katz, T Tao, A sum product estimate in finite fields and applications, Geom Funct Anal 14 (2004) 27–57.
[4] D Covert, A Iosevich, J Pakianathan, Geometric configurations in the ring of integers modulo p l, Indiana Univ Math J (2014) in press.
[5] M.Z Garaev, The sum–product estimate for large subsets of prime fields, Proc Amer Math Soc 136 (2008) 2735–2739.
[6] F Hennecart, N Hegyvári, Explicit constructions of extractors and expanders, Acta Arith 140 (2009) 233–249.
[7] F Hennecart, N Hegyvári, A note on Freiman models in Heisenberg groups, Israel J (2014) in press.
[8] M Krivelevich, B Sudakov, Pseudo-random graphs, in: Conference on Finite and Infinite Sets Budapest, in: Bolyai Society Mathematical Studies, vol X,
p 164.
[9] Hoi H Nguyen, On two-point configurations in a random set, Integers 9 (2009) 41–45.
[10] P.V Thang, L.A Vinh, Erdős–Rényi graph, Szemerédi–Trotter type theorem, and sum–product estimates over finite rings, Forum Math (2014) http://dx.doi.org/10.1515/forum-2011-0161
[11] L.A Vinh, A Szemerédi–Trotter type theorem and sum–product estimate over finite fields, Eur J Comb 32 (8) (2011) 1177–1181.