On point line incidences in vector spaces over finite fields

Discrete Applied Mathematics 177 (2014) 146–151 Contents lists available at ScienceDirect 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 article abstract info Article history: Received 31 August 2012 Received in revised form May 2014 Accepted 18 May 2014 Available online 16 June 2014 Let Fq be the finite field of q elements We show that for almost every point set P and line set L in F2q of cardinality |P | = |L| q, there exists a pair (p, l) ∈ P × L with 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 Keywords: Point-line incidences Szemerédi–Trotter theorem Residue rings Introduction Let Fq be a finite field of q elements where q is a large odd prime power Let A be 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 of A Bourgain, Katz and Tao [3] showed that when ≪ |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 > 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 F2q and one has N lines and N points in that plane for some ≪ N ≪ q2 , then there are at most O(N 3/2−ϵ ) incidences; this improves the standard bound of O(N 3/2 ) 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]) Let P be a collection of points and L be a collection of lines in F2q Then we have      {(p, l) ∈ P × L : p ∈ l} − |P ||L|  ≤ q1/2 |P ||L|   q 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 (1.1) L.A Vinh / Discrete Applied Mathematics 177 (2014) 146–151 147 In the spirit of Bourgain–Katz–Tao’s result, one can derive from Theorem 1.1 a reasonably good estimate when d = and < α < Let P be a collection of points and L be a collection of lines in F2q Suppose that |P |, |L| ≤ N = qα with + ϵ ≤ α ≤ − ϵ for some ϵ > Then we have ϵ |{(p, l) ∈ P × L : p ∈ l}| ≤ 2N − (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 set P and a line set L It follows from Theorem 1.1 that |{(p, l) ∈ P × L : p ∈ l}| ≥ |P ||L| q  − q1/2 |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 q3/2 points in F2q without any incidences Take q = p2 , with the field identified additively with F2p Note that we need to have (Fp , 0) be a subfield of Fq to 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 p3 lines For each of these lines, whenever x is in (Fp , 0), y has no values in (Fp , Bc ) Thus, there is a point set and a line set, each with about q3/2 elements, and no incidences In the case of incidences between a random point set and a random line set, we however can improve the bound q3/2 in the sense that for any α ∈ (0, 1)it suffices to take s ≥ Cα q randomly chosen lines and points in the plane to guarantee that the probability of no incidences is exponentially small α s when q is large enough Theorem 1.2 For any α > 0, there exist an integer q0 = q(α) and a number Cα > with the following property When a point set P and a line set L where |P | = |L| = s ≥ Cα q, are chosen randomly in F2q , the probability of {(p, l) ∈ P × L : p ∈ l} ≡ ∅ is at most α s , provided that q ≥ q0 Let m be a large non-prime integer and Zm be the ring of residues mod m Let γ (m) be the smallest prime divisor of m, τ (m) be the number of divisors of m, and φ(m) be Euler’s totient function We identify Zm with {0, 1, , m − 1} Define d the set of units and the set of nonunits in Zq by Z× m and Zm respectively The finite Euclidean space Zm consists of column vectors x, with jth entry xj ∈ Zm In [10], Thang and the author extended Theorem 1.1 to 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 subset E of Qd would be the same as for subsets of Zdm 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]) Let P be a collection of points and L be a collection of lines in Z2m Then we have      {(p, l) ∈ P × L : p ∈ l} − |P ||L|  ≤ 2τ (m)m |P ||L|   m φ(m)γ (m)1/2 (1.3) Notice that Theorem 1.3 is most effective when m has only few prime divisors For example, if m = pr , in the spirit of Bourgain–Katz–Tao’s result, one can obtain a reasonably good estimate when p2r −1+ϵ N p2r −ϵ Let P be a collection of α points and L be a collection of lines in Zpr Suppose that |P |, |L| N = p with 2r − + ϵ ≤ α ≤ 2r − ϵ for some ϵ > Then we have |{(p, l) ∈ P × L : p ∈ l}| ϵ N − 4r (1.4) For the lower bound, Theorem 1.3 implies that {(p, l) ∈ P × L : p ∈ l} ̸≡ ∅ when |P ||L| > 4(τ (m))2 m6 (φ(m))2 γ (m) Our next result is the finite ring analog of Theorem 1.2 More precisely, we show that for almost every point set P and line set L of cardinality |P | = |L| m, there exists a pair (p, l) ∈ P × L with p ∈ l Theorem 1.4 For any α > 0, there exist an integer m0 = m(α) and a number Cα > with the following property When a point set P and a line set L where |P | = |L| = s ≥ Cα q, are chosen randomly in Z2m , the probability of {(p, l) ∈ P × L : p ∈ l} ≡ ∅ is at most α s , provided that m ≥ m0 and γ (m) mc for some constant c > 148 L.A Vinh / Discrete Applied Mathematics 177 (2014) 146–151 Pseudo-random graphs For a graph G, let λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of its adjacency matrix The quantity λ(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 most λ It is well known (see [2, Chapter 9] for more details) that if λ is 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 vertex v of G, let N (v) denote the set of vertices of G adjacent to v and let d(v) denote its degree Similarly, for a subset U of the vertex set, let NU (v) = N (v) ∩ U and dU (v) = |NU (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 |  ≤ λ |B||C |   n 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 {Gn = G(Vn , Vn )}∞ n=1 be a sequence of bipartite graphs with |Vn | → ∞ as n → ∞ Assume that for any ϵ > 0, there exist an integer v(ϵ) and a number c (ϵ) > such that e(A, A) ≥ c (ϵ)|A|2 d¯ (Gn )/|Vn | for all |Vn | ≥ v(ϵ) and all A ⊂ Vn satisfying |A| ≥ ϵ|Vn | Then for any α > 0, there exist an integer v(α) and a number C (α) with the following property If one chooses a random subset S of Vn of cardinality s then the probability of G(S , S ) being empty is at most α s provided that s ≥ C (α)|Vn |/d¯ (G) and |Vn | ≥ v(α) To study the point-line incidences between a point set P and a line set F , we mimic the proof of [9, Lemma 8] to obtain a two-set version of the above lemma Lemma 2.3 Let {Gn = G(Vn , Un )}∞ n=1 be a sequence of bipartite graphs with |Vn | = |Un | → ∞ as n → ∞ Assume that for any ϵ > 0, there exist an integer v(ϵ) and a number c (ϵ) > such that e(A, B) ≥ c (ϵ)|A||B|d¯ (Gn )/|Vn | for all |Vn | = |Un | ≥ v(ϵ) and all A ⊂ Vn , B ⊂ Un satisfying |A||B| ≥ ϵ|Vn |2 Then for any α > 0, there exist an integer v(α) and a number C (α) with the following property If one chooses a random subset S of Vn of cardinality s and a random subset T of Un of the same cardinality s, then the probability of G(S , T ) being empty is at most α s provided that s ≥ C (α)|Vn |/d¯ (G) and |Vn | ≥ v(α) Proof We shall view S = {v1 , , vs } and T = {u1 , , us } as ordered random subsets, whose elements will be chosen in order, v1 , u1 , v2 , u2 , , vs , us For ≤ k ≤ s − 1, let Rk be the set of neighbors of the first k + chosen vertices from Vn , i.e Rk = {u ∈ Un | (vi , u) ∈ E (Gn ) for some i ≤ k + 1} and Lk be the set of neighbors of the first k chosen vertices from Un , i.e Lk = {v ∈ Vn | (v, ui ) ∈ E (Gn ) for some i ≤ k} If G(S , T ) is empty, we have vk+1 ̸∈ Lk and uk+1 ̸∈ Rk Let Ak+1 be the set of possible choices for vk+1 from Vn \{v1 , , vk } such that |Rk+1 \Rk | ≤ c (ϵ)ϵ d¯ (G), where ϵ < will be chosen small enough later and c (ϵ) is the constant from the lemma Similarly, let Bk+1 be the set of possible choices for uk+1 from Un \{u1 , , uk } such that |Lk+1 \Lk | ≤ c (ϵ)ϵ d¯ (G) We first assume that |Ak+1 ||Bk+1 | > ϵ|Vn |2 for some ≤ k ≤ s − Since |Ak+1 | ≤ |Vn |, |Bk+1 | > ϵ|Vn | Besides Bk+1 ∩ Rk = ∅, so we have e(Ak+1 , Bk+1 ) ≤ e(Ak+1 , Un \Rk ) ≤ c (ϵ)ϵ d¯ (G)|Ak+1 | < c (ϵ)|Ak+1 ||Bk+1 |d¯ (G)/|Vn | (2.1) Here, the second inequality follows from the property of the set Ak+1 that any vertex in Ak+1 has at most c (ϵ)ϵ d¯ (G) neighbors in Un \Rk On the other hand, it follows from the property of Gn that e(Ak+1 , Bk+1 ) ≥ c (ϵ)|Ak+1 ||Bk+1 |d¯ (Gn )/|Vn |, (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 |Ak+1 ||Bk+1 | ≤ ϵ|Vn |2 for ≤ k ≤ s − L.A Vinh / Discrete Applied Mathematics 177 (2014) 146–151 149 Let s be sufficiently large, for example, s ≥ 4(c (ϵ)ϵ)−1 |Vn |/d¯ (Gn ), and assume that v1 , u1 , v2 , u2 , , vs , us have been chosen Let sv be the number of vertices vk+1 that not belong to Ak+1 and su be the number of vertices uk+1 that not belong to Bk+1 We have  |Un | ≥ |Rs | ≥ |Rk+1 \Rk | ≥ sv c (ϵ)ϵ d¯ (Gn ) vk+1 ̸∈Ak+1 Hence, sv ≤ (c (ϵ)ϵ)−1 |Un |/d¯ (Gn ) ≤ s/4 Similarly, we also have su ≤ s/4 This implies that there are s − sv − su ≥ s/2 pairs (vk+1 , uk+1 ) such that vk+1 ∈ Ak+1 and uk+1 ∈ Bk+1 Since |Ak+1 ||Bk+1 | ≤ ϵ|Vn |2 for ≤ k ≤ s − 1, the number of subsets S ⊂ Vn , T ⊂ Un such that G(S , T ) is empty is bounded by   s  s  sv sv ,su ≤s/4 su 2s s/2 < |Vn | ϵ s |Vn |2(su +sv ) (ϵ|Vn |2 )s−sv −su ≤   s  s  sv ,su ≤s/4 sv su |Vn |2s ϵ s/2 s/2 < (256ϵ) |Vn |(|Vn | − 1) (|Vn | − s + 1)|Un |(|Un | − 1) (|Un | − s + 1) Taking ϵ = α /256, the lemma follows 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 − 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 , , xd ) over Fq , where two vectors are equivalent if one is a multiple of the other by an element of the field Let E R(Fdq ) 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 x1 y1 + · · · + xd yd = 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 of E R(Fdq ) is nq,d = (qd − 1)/(q − 1) and that it is dq,d -regular for dq,d = (qd−1 − 1)/(q − 1) The eigenvalues of E R(Fdq ) are easy to compute [1] Let A be the adjacency matrix of E R(Fdq ) Then, by properties of PG(q, d), A2 = AAT = µJ + (dq,d − µ)I, where µ = (qd−2 − 1)/(q − 1), J is the all one matrix and I is the identity matrix, both of size nq,d × nq,d Thus the largest eigenvalue of A is dq,d and the absolute values of  all other eigenvalues are dq,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 graph E R(Fdq ) is an  qd − qd−1 − q−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 Zm as follows For any x ∈ Zdm \(Z0m )d , we denote [x] the equivalence class of x in Zdm \(Z0m )d , where x, y ∈ Zdm \(Z0m )d are equivalent d if and only if x = ty for some t ∈ Z× m Let E R (Zm ) 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 = In [10], Thang and the author obtained the following pseudo-randomness of the Erdős–Rényi graph E R(Zdm ) Lemma 3.2 ([10, Theorem 2.4]) For any m, d ≥ 2, the Erdős–Rényi graph E R(Zdm ) is an  md − (m − φ(m))d md−1 − (m − φ(m))d−1 φ(m) , φ(m) , 2τ (m)md−1 φ(m)γ (m)(d−2)/2  − graph In order to prove Lemma 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 graph ZP m,d is defined as follows The vertex set of the zero-product graph ZP m,d is the set V (ZP m,d ) = Zdm \(Z0m )d Two vertices a and b ∈ V (ZP m,d ) are connected by an edge, (a, b) ∈ E (ZP m,d ), if and only if a · b = Thang and the author proved that for any m, d ≥ 2, the zero-product graph ZP m,d is an  md − (m − φ(m))d , md−1 − (m − φ(m))d−1 , 2τ (m)md−1 γ (m)(d−2)/2  − graph (3.1) 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 size φ(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 in Lemma 3.2 can 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) 150 L.A Vinh / Discrete Applied Mathematics 177 (2014) 146–151 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 Let Gq,3 be 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 = x1 y1 + · · · xd yd = 0} Note that the graph Gq,3 is just a bipartite version of the Erdős–Rényi graph in Section 3.1 It follows that Gq,3 is a regular bipartite graph of valency d¯ (Gq,3 ) = (q2 − 1)/(q − 1) Besides, from Lemmas 2.1 and 3.1, for any E , F ⊂ PG(q, 3), we have      e(E , F ) − q − |E ||F | ≤ q1/2 |E ||F |   q −1 (4.1) We can embed the plane F2q into P F3q by identifying (x, y) with the equivalence class of (x, y, 1) Any line in F2q also can be represented uniquely as an equivalence class in P F3q of some non-zero element h ∈ F3q For each x ∈ F3q , we denote [x] the equivalence class of x in P F3q The relation x · y = x1 y1 + x2 y2 + x3 y3 = 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 F2q can be interpreted as the number of edges between two vertex sets of Gq,3 For any ϵ > 0, |E ||F | ≥ ϵ q4 and q ≥ 2/ϵ , it follows from (4.1) that e(E , F ) ≥ q2 − q3 −1  |E ||F | − q1/2 |E ||F | > q2 − 2( q3 − 1) |E ||F | = d¯ (Gq,d ) 2|V (Gq,d )| |E ||F | Let c (ϵ) = 1/2 and v(ϵ) ≥ 8/ϵ , Theorem 1.2 now follows from Lemma 2.3 Notice that, Theorem 1.1 also 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 Let Pm,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 = x1 y1 + · · · xd yd = 0} Note that the graph Pq,3 is just a bipartite version of the Erdős–Rényi graph in Section 3.2 It follows that Pq,3 is a regular bipartite graph of valency d¯ (Pq,3 ) = m2 − (m − φ(m))2 φ(m) We can embed the plane Z2m into P Z3m by identifying (x, y) with the equivalence class of (x, y, 1) Any line in Z2m also can be represented uniquely as an equivalence class in P Z3m of some non-zero element h ∈ Z3m For each x ∈ Z3m , we denote [x] the equivalence class of x in P Z3m The relation x · y = x1 y1 + x2 y2 + x3 y3 = 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 Z2m can be interpreted as the number of edges between two vertex sets of Pm,3 From Lemmas 2.1 and 3.2, for any E , F ⊂ PG(m, 3) we have   2    e(E , F ) − m − (m − φ(m)) |E ||F | ≤ 2τ (m)m |E ||F |   3 / m − (m − φ(m)) φ(m)γ (m) Hence, it is easy to check that, for any ϵ > 0, if |E ||F | (φ(m))2 γ (m) 16 > , m2 (τ (m))2 ϵ (4.2) ϵ m4 and (4.3) 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) and γ (m) mc for some constant c > then Eq (4.3) holds, Theorem 1.4 now follows from Lemma 2.3 Notice that, Theorem 1.3 also follows immediately from Eq (4.2) L.A Vinh / Discrete Applied Mathematics 177 (2014) 146–151 151 Acknowledgment This research was supported by Vietnam National University- Hanoi project QGTD.13.02 References [1] [2] [3] [4] [5] [6] [7] [8] N Alon, M Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs Combin 13 (1997) 217–225 N Alon, J.H Spencer, The Probabilistic Method, second ed., Willey-Interscience, 2000 J Bourgain, N Katz, T Tao, A sum product estimate in finite fields and applications, Geom Funct Anal 14 (2004) 27–57 D Covert, A Iosevich, J Pakianathan, Geometric configurations in the ring of integers modulo pl , Indiana Univ Math J (2014) in press M.Z Garaev, The sum–product estimate for large subsets of prime fields, Proc Amer Math Soc 136 (2008) 2735–2739 F Hennecart, N Hegyvári, Explicit constructions of extractors and expanders, Acta Arith 140 (2009) 233–249 F Hennecart, N Hegyvári, A note on Freiman models in Heisenberg groups, Israel J (2014) in press 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 (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  ... proved the following theorem on point- line incidences in vector spaces over finite rings Theorem 1.3 ([10, Theorem 1.6]) Let P be a collection of points and L be a collection of lines in Z2m Then... 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... no values in (Fp , Bc ) Thus, there is a point set and a line set, each with about q3/2 elements, and no incidences In the case of incidences between a random point set and a random line set,