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 D O D UY H IEU & L E A NH V INH A BSTRACT Let Fq be a finite field of q elements We show that if A ⊂ Fq of cardinality |A| ≳ q1/2 , then |A|2 |(A − A) · (A − A) · · (A − A)| ≳ q, 1/2n−1 q , where the product is taken n times We also obtain similar results in the setting of finite cyclic rings I NTRODUCTION The classical Erd˝os distance problem asks for the minimal number of distinct distances determined by a finite point set in Rn , n ≥ 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 least cN/ log N 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 = pr (r ≥ 1) where p is a sufficiently large prime and r 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 > that is independent of X and Y , such that X ≤ CY Let Fq denote a finite field with q elements For E ⊂ Fnq (n ≥ 2), the finite analogue of the classical Erd˝os distance problem is to determine the smallest possible cardinality of the set ∆Fq (E) = { x − y = (x1 − y1 )2 + · · · + (xn − yn )2 : x, y ∈ E} ⊂ Fq 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 (2016) 2126 D O D UY H IEU & L E A NH V INH prime, q ≡ (mod 4), then for every ε > and E ⊂ F2q with |E| ≤ Cε q2−ε , there exists δ > such that |∆Fq (E)| ≥ Cδ |E|1/2+δ for some constants Cε , Cδ Iosevich and Rudnev ([9]) used Fourier analytic methods to show that there are absolute constants c1 , c2 > such that, for any odd prime power q and any set E ⊂ Fdq of cardinality |E| ≥ c1 qn/2 , we have (1.1) |∆Fq (E)| ≥ c2 min{q, q(n−1)/2 |E|} Iosevich and Rudnev ([9]) reformulated the question in analogy with the Falconer distance problem: how large does E ⊂ Fnq , 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 rings Zq = Z/qZ (q = pr , p is a sufficiently large prime) Here, the distance set of E ⊂ Zdq is defined similarly by ∆Zq (E) = { x − y = (x1 − y1 )2 + · · · + (xn − yn )2 : x, y ∈ E} ⊂ Zq In [5], Covert, Iosevich, and Pakianathan obtained a nearly sharp bound for the distance problem in Zdq More precisely, they proved that if E ⊂ Znq of cardinality |E| ≳ r (r + 1)q(2r −1)n/(2r )+1/(2r ) , then Z× q ⊂ ∆Zq (E) In [17], the second listed author extended this result using graph-theoretic methods, and showed, roughly speaking, that any sufficiently large subset E ⊂ Znq determines all possible non-degenerate k-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 ⊆ Fq , 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) = Fq , where the product is taken n times Denote Vn (A) = (A − A) · (A − A) · · (A − A), where the product is taken n times Balog ([2]) obtained the following results On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2127 k Theorem 1.2 ([2, Theorem 1.1]) Let A ⊆ Fq with |A| ≥ q1/2+1/2 , where k > We have V2k+1 (A) = Fq Corollary 1.3 ([2, Corollary 1]) Let A ⊆ Fq with |A| > q1/2 We have k |Vk (A)| ≥ q1−1/2 Using the graph theoretic method and Corollary 1.3, we obtain the following theorem Theorem 1.4 Let A ⊆ Fq of cardinality |A| ≳ q1/2 We have |A|2 |Vn (A)| ≳ q, 1/2n−1 q n Specifically, Theorem 1.4 implies that if A ⊆ Fq of cardinality |A| ≳ q1/2+1/2 , then |Vn (A)| ≳ q Similarly, we also prove the following result over finite rings Theorem 1.5 Let A ⊆ Zq such that |A| ≳ q1−1/(2r ) ; then, |Vn (A)| ≳ p r , |A|2 n−1 r p r −1+1/2 Using the graph-theoretic method and Theorem 1.4, we obtain the following improvement of Theorem 1.2 k−1 Theorem 1.6 Let A ⊆ Fq with |A| ≳ q1/2+1/(3·2 ) , where k > We have V2k+1 (A) = Fq Similarly, we also prove the following result over finite rings Theorem 1.7 Let A ⊆ Zq , a finite ring with q elements Suppose that k−1 |A| ≳ 2r q1−1/(2r )+1/(3r ·2 ) ; then, Z×q ⊆ V2k+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 S OME L EMMAS In this section, we follow closely [2, Section 2] to extend several sum-product type results to the setting of cyclic rings Let Zq = Z/qZ (q = pr , r is a sufficiently large prime) be a finite cyclic ring of order q Denote Z×q the set of units of Zq and Z0q = Zq \ Z× q Lemma 2.1 Let A, B ⊆ Zq with |A| |B| > q2−1/r We have Z× q ⊆ A−A (B − B) \ Z0q 2128 D O D UY H IEU & L E A NH V INH Proof For any x ∈ Z×q , consider the set A − xB ⊆ Zq Since |A − xB| < p r < |A| |B| , p r −1 ❐ there exist pr −1 different pairs (a1 , b1 ), (a2 , b2 ), , (apr −1 , bpr −1 ) such that − xbi = aj − xbj , or x(bi − bj ) = − aj If = aj , then bi = bj , so aj = aj with i = j Let M = {a1 , a2 , , apr −1 } Since |M − M| > |M| = pr −1 , there are , aj such that − aj ∈ Z×q , or x = (bi − bj )/(ai − aj ) 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 | |V W | From Lemmas 2.1 and 2.2 we prove the following result over finite rings n Lemma 2.3 Let A ⊆ Zq with |A| ≥ q1−1/(2r ) We have |Vn (A)| ≥ q1−1/(r ) Proof Inserting U = V = (A−A)\Z0q , W = V n , and G = Zq into Lemma 2.2, it follows from Lemma 2.1 that if |A| > pr −1/2 , then |UV −1 | ≥ pr − pr −1 Therefore, and by induction we get (p r − p r −1 )|V n | ≤ |V n+1 |2 , n −1 (p r − p r −1 )2 n |V | ≤ |V n+1 |2 n |Vn (A)| ≥ |V n | ≥ cq1−1/(r ) ❐ Finally, using the trivial bounds |V | = |(A − A) \ Z0q | > pr −1/2 − pr −1 , we have P RODUCT-S UM G RAPHS 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 ([1, Chapter 9]) that if λ is much smaller than the degree d, then G has 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 On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2129 of ordered pairs (u, w) such that u ∈ U , w ∈ W , and (u, w) is an edge of G Then, we have e(B, C) − d|B| |C| n ≤ λ |B| |C| For any δ ∈ F×q , the product-sum graph PSq (δ) is defined as follows The vertex set of the product graph PSq (δ) is the set F∗q × Fq Also, the two vertices a = (a1 , a2 ) and b = (b1 , b2 ) ∈ V (PSq (δ)) are connected by an edge, {a, b} ∈ E(PSq (λ)), if and only if a1 b1 (a2 + b2 ) = δ In [16], the second listed author proved a pseudo-randomness of the product graph PSq (δ) Note that, in the statement of [16, Theorem 3.6], all but the largest eigenvalue of the graph PSq (δ) are bounded by 3q However, looking carefully at the “error graph” E in the proof of that theorem, we can bound all but the largest eigenvalue of E by √ 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 graph PSq (δ) is a ((q − 1)q, q − 1, 2q )-graph In this paper, we will study an extension of the above theorem to the setting of finite cyclic rings Suppose that q = pr for a sufficiently large prime p For any δ ∈ Z×q , the product-sum graph PSRq (δ) is defined as follows The vertex set of the sum-product graph PSRq (δ) is the set V (PSRq (δ)) = Z×q × Zq Two vertices a = (a1 , a2 ) and b = (b1 , b2 ) ∈ V (PSRq (δ)) are connected by an edge in E(PSRq (δ)), if and only if a1 b1 (a2 + b2 ) = δ We have the following pseudo-randomness of the product-sum graph PSRq (δ) Theorem 3.3 For any δ ∈ Z×q , the product-sum graph PSRq (δ) is a (p 2r − p 2r −1 , p r − p r −1 , 2r p 2r −1 )-graph Proof It is easy to see that PSRq (δ) is a regular graph of order p2r − p2r −1 and valency pr − pr −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 (PSRq (δ)), we count the number of paths of length two from a to b More precisely, we count the number of solutions of the system (3.1) a1 x1 (a2 + x2 ) = δ, b1 x1 (b2 + x2 ) = δ, (x1 , x2 ) ∈ Z× p r × Zp r For each solution x1 ∈ Z×pr of the equation (3.2) x1 (a2 − b2 ) = δ δ − , a1 b1 there exists a unique x2 ∈ Zpr satisfying the system of equations (3.1) Therefore, we only need to count the number of solutions of equation (3.2) 2130 D O D UY H IEU & L E A NH V INH For any prime p, the p-adic valuation vp (n) of an nonzero integer n is the highest power of p that divides n Let α = vp (a2 − b2 ); then, clearly we have ≤ α ≤ r − (otherwise, α = r ; then, a2 = b2 and a1 = b1 ) If we have vp (δ/a1 − δ/b1 ) = α, then it is easy to see that equation 3.2 has no solution Suppose that vp (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 solution x1 ∈ Zpr −α of βx1 = γ Putting this back in to equation (3.2) gives us pα solutions Hence, the system of equation (3.1) has p α solutions if vp (δ/a1 − δ/b1 ) = α, and no solution otherwise Therefore, suppose that we have any two distinct vertices a = (a1 , a2 ) and b = (b1 , b2 ) ∈ V (PSRq (δ)) Let also α = vp (a2 − b2 ); then, (a1 , a2 ) and (b1 , b2 ) have p α common neighbors if vp (δ/a1 − δ/b1 ) = α, and no common neighbor otherwise Let A be the adjacency matrix of PSRq (δ) It follows that (3.3) A2 = J + (p r − p r −1 − 1)I − r −1 Eα + α=0 r −1 (p α − 1)Fα , α=1 where we have the following: • J is the all-one matrix • I is the identity matrix • Eα is the adjacency matrix of the graph BE,α , where the vertex set of BE,α is Z×q × Zq Furthermore, for any two vertices U = (a, b) and V = (c, d) ∈ V (BE,α ), (U, V ) is an edge of BE,α if and only if α = vp (a2 − b2 ) = vp (δ/a1 − δ/b1 ) • Fα is the adjacency matrix of the graph BF,α , where the vertex set of BF,α is Z×q × Zq Furthermore, for any two vertices U = (a, b) and V = (c, d) ∈ V (BF,α ), (U, V ) is an edge of BF,α if and only if α = vp (a2 − b2 ) = vp (δ/a1 − δ/b1 ) For any α > 0, BE,α is a regular graph of order less than (p r −α − p r −α−1 )(p r − p r −α + p r −α−1 ), and BF,α is a regular graph of order less than (pr −α − pr −α−1 )2 Hence, all eigenvalues of Eα are at most (pr −α − pr −α−1 )(pr − pr −α + pr −α−1 ), and all eigenvalues of Fα are at most (pr −α − pr −α−1 )2 Note that E0 is a zero matrix Since PSRq (δ) is a (pr − pr −1 )-regular graph, pr − pr −1 is an eigenvalue of A with the all-one eigenvector The graph PSRq (δ) is connected; therefore, the eigenvalue pr − pr −1 has multiplicity one Since the graph PSRq (δ) contains (many) triangles, it is not bipartite Hence, |θ| < pr − pr −1 for any other eigenvalue θ Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1⊥ , On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2131 so Jv θ =0 It follows from (3.3) that (θ − p r + p r −1 + 1)vθ = − r −1 α=0 Eα + r −1 (p α − 1)Fα vθ α=1 Hence, v θ is also an eigenvalue of − r −1 α=0 Eα + r −1 α=1 (p α − 1)Fα Since eigenvalues of the sum of the matrices are bounded by the sum of the largest eigenvalues of the summands, we have θ ≤ p r − p r −1 − + + r −1 α=1 r −1 (p r −α − p r −α−1 )(p r − p r −α + p r −α−1 ) α=0 (p α − 1)(p r −α − p r −α−1 )2 ❐ This implies that θ2 ≤ 2r p2r −1 , and the lemma follows P ROOF OF T HEOREM 1.4 AND T HEOREM 1.6 4.1 Proof of Theorem 1.4 As a consequence of Theorem 3.2, we have the following lemma Lemma 4.1 For any A ⊆ F×q , B, C ⊆ Fq with |B| > 1, we have |{a(b − c) : a ∈ A, b ∈ B, c ∈ C}| ≳ q, |A| |B| |C| q Proof Let D = {a(b − c) : a ∈ A, b ∈ B, c ∈ C} ∩ F×q , and let N be the number of solutions of equation ad(b − c) = 1, (a, b, c, d) ∈ A × B × C × D −1 It is clear that N = |A| |B| |C| − |A| |B ∩ C| Besides, N is the number of edges between two vertex sets A × B and D −1 × (−C) of the product-sum graph PSq From Lemma 3.1 and Theorem 3.2, we have |A| |B| |C| − |A| |B ∩ C| − |A| |B| |C| |D| q ≤ 2q|A| |B| |C| |D|, or equivalently, |A| |B| |C| − |A| |B ∩ C| ≤ |A| |B| |C| |D| + 2q|A| |B| |C| |D| q D O D UY H IEU & L E A NH V INH 2132 Let t = |D| ≥ 0; then, |A| |B| |C| t + 2q t − |A| |B| |C| + q |A| |B ∩ C| ≥ 0, |B| |C| which implies that |D| ≥ ≥ = − 2q + 2q + 4|A| |B| |C|/q − 4|A| |B ∩ C|/q |A| |B| |C|/q − 2q + 2q + |A| |B| |C|/q |A| |B| |C|/q 2−1 |A| |B| |C| 2q + 2q + |A| |B| |C|/q ≳ q, |A| |B| |C| q , ❐ where the third line follows from the fact that |B| |C| ≥ 2|C| > 43 |B ∩ C| This concludes the proof of the lemma We are now ready to give a proof of Theorem 1.4 Putting A = Vn−1 (A), B = C = A into Lemma 4.1, and using Corollary 1.3, we have |A|2 |Vn (A)| ≳ q, 1/2n−1 q , 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 ⊆ Fq , such that |A| |B| |C| |D| ≥ 2q3 We have AB(C − D) = Fq Proof Let Hδ = {(a, b, c, d) ∈ A × B × C × D : ab(c − d) = δ} for any δ ∈ F∗ q Then, |Hδ | is the number of edges between two vertex sets A × C and B × D of the product-sum graph PSq (δ) From Lemma 3.1 and Theorem 3.2, we have |Hδ | − |A| |B| |C| |D| q |Hδ | ≥ |A| |B| |C| |D| − 2q|A| |B| |C| |D| q ≤ 2q|A| |B| |C| |D|, or equivalently, ❐ Therefore, if |A| |B| |C| |D| > 2q3 , then |Hδ | > This implies AB(C −D) = Fq , and this completes the proof of Lemma 4.2 On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2133 We are now ready to give a proof of Theorem 1.6 Putting A = Vk (A), B = Vk (A), C = D = A into Lemma 4.2, and using Theorem 1.4, we have that if k−1 |A| ≥ cq1/2+1/(3·2 ) , then V2k+1 (A) = Fq , concluding the proof of the theorem P ROOF OF T HEOREM 1.5 AND T HEOREM 1.7 5.1 Proof of Theorem 1.5 Lemma 5.1 Let q = pr with p an odd prime and r ≥ an integer For any A, B, C ⊆ Zq such that |B| > 4|Z0q |/3, we have |a(b − c) : a ∈ A, b ∈ B, c ∈ C| ≳ p r , |A| |B| |C| 2r p2r −1 Proof Let D = {a(b − c) : a ∈ A, b ∈ B, c ∈ C} ∩ Z×q , and let N be the number of solutions of equation ad(b − c) = 1, (a, b, c, d) ∈ A × B × C × D −1 It is clear that N = |A| |B| |C| − |A| t∈Zq |B ∩ (C + t)| Moreover, N is the number of edges between two vertex sets A×B and D −1 ×(−C) of the product-sum graph PSRq (δ) From Lemma 3.1 and Theorem 3.3, we get |A| |B| |C| − |A| t∈Z0q |B ∩ (C + t)| − |A| |B| |C| |D| pr ≤ 2r p 2r −1 |A| |B| |C| |D|, or equivalently, |A| |B| |C| |D| + 2r p 2r −1 |D| pr − |A| |B| |C| + |A| |B| |C| t∈Z0q |B ∩ (C + t)| ≥ Let x = |D| ≥ 0; then, |A| |B| |C| x + 2r p 2r −1 x pr − |A| |B| |C| + |A| |B| |C| t∈Z0q |B ∩ (C + t)| ≥ 0, D O D UY H IEU & L E A NH V INH 2134 which implies that |D| ≥ − 2r p 2r −1 |A| |B| |C| p−r 2r p2r −1 + 4|A| |B| |C|p−r − 4|A| + ≥ = t∈Zq |B ∩ (C + t)|p −r |A| |B| |C| p−r − 2r p 2r −1 + 2r p 2r −1 + |A| |B| |C|p −r |A| |B| |C| p−r 2r p2r −1 2( ≳ pr , |A| |B| |C| + 2r p 2r −1 + |A| |B| |C| p −r ) |A| |B| |C| 2r p2r −1 , ❐ where the third line follows from the fact that |B| |C| > 4|Z0q | |C| ≥ 4|B ∩ C| This concludes the proof of the lemma We are now ready to give a proof of Theorem 1.5 Putting A = Vk (A), B = C = A into Lemma 5.1, and using Lemma 2.3, we have |Vn (A)| ≳ p r , |A|2 n−1 r p r −1+1/2 , 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 ⊆ Zq , such that |A| |B| |C| |D| ≥ 2r q4−1/r We have Z×q ⊆ AB(C − D) Proof Let Hδ = {(a, b, c, d) ∈ A × B × C × D : ab(c − d) = δ} for any δ ∈ Z× q Then, |Hδ | is the number of edges between two vertex sets A × C and B × D of the product-sum graph PSRq (δ) From Lemma 3.1 and Theorem 3.3, we have |Hδ | − |A| |B| |C| |D| pr ≤ 2r p 2r −1 |A| |B| |C| |D|, or equivalently, |Hδ | ≥ |A| |B| |C| |D| − 2r p 2r −1 |A| |B| |C| |D| pr On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2135 Therefore, if |A| |B| |C| |D| > 2r p4r −1 , then |Hδ | > This implies that and this completes the proof of Lemma 5.2 ❐ Z× q ⊆ AB(C − D), We are now ready to give a proof of Theorem 1.7 Putting A = Vk (A), B = Vk (A), C = D = A into Lemma 5.2, and using Theorem 1.5, we have Z× q ⊆ V2k+1 (A) √ k−1 if |A| ≥ c 2r q1−1/(2r )+1/(3r ·2 ) This concludes the proof of the theorem Acknowledgements This research was supported by Vietnam National University, Hanoi Project QGTD.13.02 R EFERENCES [1] N A LON and J H S PENCER, The Probabilistic Method, 2nd ed., Wiley-Interscience Series in Discrete 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Putting A = Vk (A), B = Vk (A), C = D = A into... For two (not necessarily) disjoint subsets of vertices U, W ⊂ V , let e(U, W ) be the number On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2129 of ordered pairs (u, w) such that... |C| |D| pr On the Volume Set of Boxes in Vector Spaces Over Finite Fields 2135 Therefore, if |A| |B| |C| |D| > 2r p4r −1 , then |Hδ | > This implies that and this completes the proof of Lemma 5.2