On three variable expanders over finite fields

March 1, 2014 9:43 WSPC/S1793-0421 203-IJNT 1350118 International Journal of Number Theory Vol 10, No (2014) 689–703 c World Scientific Publishing Company DOI: 10.1142/S1793042113501182 ON THREE-VARIABLE EXPANDERS OVER FINITE FIELDS Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only LE ANH VINH University of Education Vietnam National University Hanoi, Vietnam vinhla@vnu.edu.vn Received 26 February 2013 Accepted 15 September 2013 Published December 2013 Let Fq be the finite field with q elements and P (x, y) be a polynomial in Fq [x, y] Using additive character sum estimates, we study expander property of the function x1 + P (x2 , x3 ) We give an alternative proof using spectra of sum–product graphs in the case of P (x, y) = xy, and also extend the problem in the setting of finite cyclic rings Keywords: Expanders; extractors; spectral graphs; finite fields Mathematics Subject Classification 2010: 11L40, 11T30, 11E39 Introduction Let Fq be the finite field with q elements, where q is an odd prime power, and let E be a finite subset of Fdq , the d-dimensional vector space over Fq Given a function f : Fdq → Fq , define f (E) = {f (x) : x ∈ E}, the image of f under the subset E We say that f is a d-variable expander with expansion index if |f (E)| ≥ C |E|1/d+ for every subset E, possibly under some general density or structural assumptions on E The question of whether certain polynomials have the expander property has been studied in various classical probos distance problem [6] lems For example, given a finite subset E ⊂ Rd , the Erd˝ d d deals with the function ∆ : R × R → R where ∆(x, y) = x− y It is conjectured that ∆ is a 2d-variable expander with expansion index 1/2d, i.e |∆(E, E)| |E|2/d This problem in the Euclidean plane has recently been solved by Guth and Katz [13] 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 [22, 29], and the references contained therein Here and throughout, X Y means that X ≥ CY for some large constant C and X Y means that Y = o(X) 689 March 1, 2014 9:43 WSPC/S1793-0421 690 203-IJNT 1350118 L A Vinh Let Fq denote a finite field with q elements, where q, a power of an odd prime, is viewed as an asymptotic parameter For E ⊂ Flq (l ≥ 2), the finite analogue of the classical Erd˝os distance problem is to determine the smallest possible cardinality of the set Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only ∆(E) = { x − y = (x1 − y1 )2 + · · · + (xl − yl )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 prime, q ≡ (mod 4), then for every ε > and E ⊂ F2q with |E| ≤ Cε q , there exists δ > such that |∆(E)| ≥ Cδ |E| +δ for some constants Cε , Cδ The relationship between ε and δ in their arguments, however, is difficult to determine In addition, it is quite subtle to go up to higher-dimensional cases with these arguments Iosevich and Rudnev [20] 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 ⊂ Fdl of cardinality |E| ≥ c1 q l/2 , we have |∆(E)| ≥ c min{q, q l−1 |E|} (1.1) In [33], Vu gave another proof of (1.1) using the graph-theoretic method (see also [30] for a similar proof) Iosevich and Rudnev reformulated the question in analogy with the Falconer distance problem: how large does E ⊂ Flq , l ≥ 2, needed to be ensure that ∆(E) contains a positive proportion of the elements of Fq The above l+1 result implies that if |E| ≥ 2q then ∆(E) = Fq directly in line with Falconer’s result in Euclidean setting that for a set E 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 improvable, in line with the Falconer distance conjecture described above However, Hart, Iosevich, Koh and Rudnev discovered in [17] that the arithmetic of the problem makes the exponent (l + 1)/2 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 [4], Chapman et al took a first step in this direction by showing that if E ⊂ F2q satisfies |E| ≥ q 4/3 then |∆(E)| ≥ cq This is in line with Wolff’s result for the Falconer conjecture 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 non-expanding then it may imply that the 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 and Szemer´edi [7] conjectured that for any subset A ⊂ Z, either |A + A| or |A · A| is large, that is max(|A + A|, |A · A|) |A|2 March 1, 2014 9:43 WSPC/S1793-0421 203-IJNT 1350118 On Three-Variable Expanders Over Finite Fields 691 To support this conjecture, Erd˝ os and Szemer´edi only gave the bound Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only max(|A + A|, |A · A|) |A|1+ for a positive constant > Explicit bounds on have been studied by many researchers (see [5] and the references therein) The current best known bound ≥ 3/14 − δ where δ → as |A| → ∞ is due to Solymosi [26] These bounds hold in the more general context of finite subsets of R In this case, the best known bound, due to Solymosi [28], is ≥ 1/3 The sum–product problems have also been explored in the context of the finite fields Fq In this context, one may not rely on the topological structure of the real spaces The first non-trivial result is due to Bourgain, Katz and Tao [3] If A ⊂ Fp , p is a prime, and if |A| ≤ p1− for some > 0, then there exists δ > such that then 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, [8, 9, 18, 19, 27, 31, 33] and the references therein A related question that has recently received attention is the following Let A ⊂ Fq , how large does A need to be to make sure that F∗q ⊂ dA2 = AA + · · · + AA(d times) Bourgain [2] showed that if A ⊂ Fq of cardinality |A| ≥ Cq 3/4 then A · A + A · A + A · A = Fq Glibichuk [10] proved in the case of prime fields Fp that for d = 8, one can take √ |A| > p, and extended [11] this result to arbitrary finite fields under a weaker assumption Glibichuk and Konyagin [12] proved that if A is a subgroup of F∗p (p is a large prime), and |A| > pδ , δ > 0, then kA2 = Zp with k 41/δ In [16, 17], a geometric approach to this problem has been developed In particular, it was proved that if |A| > q 1/2+1/2d then F∗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´ arkăozy [23, 24] also studied the expanding property of the set |2A | 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 = Fq whenever |A| q 3/4 These results imply that f1 (x1 , x2 , x3 , x4 ) = x1 x2 + x3 x4 and f2 (x1 , x2 , x3 , x4 ) = x1 + x2 + x3 x4 are four-variable expanders for |A| q 2/3 Furthermore, Garaev [9] considered these functions over some special sets A, B, C, D to obtain new results on the sum–product problem in finite fields The author reproved these results using graph theory methods in [32] Using bounds of multiplicative character sums, Shparlinski [25] extended the class of sets which satisfy this property Shparlinski also asked for the size of the set |A + B · C| for large subsets A, B, C ⊂ Fq More precisely, he proved that q − |A + B · C| q3 |A||B||C| March 1, 2014 9:43 WSPC/S1793-0421 692 203-IJNT 1350118 L A Vinh Note that, this result is only effective when |A||B||C| q In prime fields, it follows from the result of Glibichuck and Konyagin [12] that for |A| < q 1/2 one has that |A + A · A| |A|7/6 This shows that under these constraints one has that f (x1 , x2 , x3 ) = x1 + x2 x3 is a three-variable expander of expansion index 1/18 Our first result is that f (x1 , x2 , x3 ) = x1 + x2 x3 is also a three-variable expander when |A| p1/2 More precisely, we have the following theorem Theorem 1.1 For any subsets A, B, C ⊆ Fq , we have Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only |A + B · C| q, |A||B||C| q In [14, 15], Gyarmati and S´ arkăozy studied solvability of the general equations a + b = f (c, d) and ab = f (c, d), a ∈ A, b ∈ B, c ∈ C, d ∈ D where A, B, C, D ⊂ Fq and f (x, y) ∈ Fq [x, y] We will use these results to study the expanding property of the polynomial x1 + f (x2 , x3 ) where f (x, y) ∈ Fq [x, y] Our next result is the following theorem Theorem 1.2 Assume that f (x, y) ∈ Fq [x, y] is not of the form g(x) + h(y) with g(x), h(x) ∈ Fq [x] Write f (x, y) in the form n gk (y)xk , f (x, y) = (1.2) k=0 with gk (y) ∈ Fq [y], and let K denote the greatest k value with the property that gk (y) is not identically constant : gK (y) ≡ c and either K = n or gK+1 , , gn (y) are identically constant Denote the degree of the polynomial gK (y) by D and assume that (K, q) = (note that K, D > by the assumption) For any subsets A, B, C ⊆ Fq , we have |A + f (B, C)| q, |A||B||C| q(D + (K − 1)q 1/2 ) Although Theorem 1.1 can be obtained directly from Theorem 1.2 by setting f (x, y) = xy, we choose to present a graph-theoretic proof to show how different techniques can be used to deal with problems of this kind Using the same techniques, we extend this result in the setting of the finite cyclic ring Let m be a large integer and Zm = Z/mZ be the ring of residues mod m Let γ(m) be the smallest prime divisor of m and τ (m) be the number of divisors of m We identify Zm with {0, 1, , m − 1} Define the set of units and the set of non-units in Zm by Z× m and Z0m , respectively We have the following finite ring analogue of Theorem 1.1 Theorem 1.3 For any subsets A, B, C ⊆ Zm , we have |A + B · C| m, |A||B||C|γ(m) (τ (m))1/2 m2 March 1, 2014 9:43 WSPC/S1793-0421 203-IJNT 1350118 On Three-Variable Expanders Over Finite Fields 693 Note that Theorem 1.3 is most effective when m has only few prime divisors and γ(m) m1/ for some > In the next section, we will prove Theorem 1.2 using additive character sum estimates Finally, we give graph-theoretic proofs of Theorems 1.1 and 1.3 in Sec General Three-Variable Expanders Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only We will prove Theorem 1.2 in this section Our first tool is the following character sum estimate proved by Gyarmati and S arkăozy in [14] Lemma 2.1 ([14, Theorem C]) Assume that α(x), β(x) are complex-valued functions on Fq , ψ is a non-trivial additive character of Fq , f (x, y) ∈ Fq [x, y], and f (x, y) is not of the form g(x) + h(y) with g(x), h(x) ∈ Fq [x] Write f (x, y) in the form n gk (y)xk , f (x, y) = (2.1) k=0 with gk (y) ∈ Fq [y], and let K denote the greatest k value with the property that gk (y) is not identically constant : gK (y) ≡ c and either K = n or gK+1 , , gn (y) are identically constant Denote the degree of the polynomial gK (y) by D and assume that (K, q) = (note that K, D > by the assumption) Write α(x)β(y)ψ(f (x, y)), S= x∈Fq y∈Fq |α(x)|2 X= x∈Fq and |β(y)|2 Y = y∈Fq Then we have |S| ≤ (XY q(D + (K − 1)q 1/2 ))1/2 Using [14, Theorem C], Gyarmati and S arkăozy [15, Theorem 3] studied the solvability of equation a + b = f (c + d), a ∈ A, b ∈ B, c ∈ C, d ∈ D, where A, B, C, D ⊆ Fq We mimic their arguments to study the equation a + f (b, c) − a1 − f (b1 , c1 ) = 0, a ∈ A, b ∈ B, c ∈ C, a1 ∈ A1 , b1 ∈ B1 , c1 ∈ C1 , (2.2) where A, B, C, A1 , B1 , C1 ⊂ Fq Lemma 2.2 Assume that q is a prime power, f (x, y) ∈ Fq [x, y], f (x, y) is not of the form g(x) + h(y) Define K and D as in Lemma 2.1, and assume that March 1, 2014 9:43 WSPC/S1793-0421 694 203-IJNT 1350118 L A Vinh (K, q) = If A, B, C, A1 , B1 , C1 ⊂ Fq , and the number of solutions of Eq (2.2) is denoted by N, then we have N− |A||B||C||A1 ||B1 ||C1 | ≤ q(D + (K − 1)q 1/2 ) |A||B||C||A1 ||B1 ||C1 | q (2.3) Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only Proof We first recall some character sum estimates over the finite field Fq Let Ψ be the set of all additive characters of Fq , let ψ0 be the principal character, and let Ψ∗ ⊂ Ψ be the set of all non-principal characters We have the following identities: ψ(z) = ψ∈Ψ q z = 0, otherwise, q ψ = ψ0 , (2.4) and ψ(z) = (2.5) otherwise z∈Fq 2πia Note that if the field is Zp , then the characters are just e p and the identity follows by summing up the geometric series For more information about the additive characters, we refer the reader to [21, Sec 11.1] Therefore, we have N= q ψ(a − a1 + f (b, c) − f (b1 , c1 )) a∈A,b∈B,c∈C ψ∈Ψ a1 ∈A1 ,b1 ∈B1 ,c1 ∈C1 Separating the principle character term ψ = ψ0 , we obtain N = |A||B||C||A1 ||B1 ||C1 | q + q ψ(a − a1 + f (b, c) − f (b1 , c1 )), ψ∈Ψ∗ a∈A,b∈B,c∈C a1 ∈A1 ,b1 ∈B1 ,c1 ∈C1 which implies that N− |A||B||C||A1 ||B1 ||C1 | q = q ψ(a) ψ∈Ψ∗  a∈A a1 ∈A1   ¯ 1)  ψ(a ψ(f (a, b))  ¯ (a1 , b1 )) ψ(f × b1 ∈B1 ,c1 ∈C1 b∈B,c∈C March 1, 2014 9:43 WSPC/S1793-0421 203-IJNT 1350118 On Three-Variable Expanders Over Finite Fields 695 Hence, N− ≤ |A||B||C||A1 ||B1 ||C1 | q q ¯ 1) ψ(a ψ(a) ψ∈Ψ∗ a∈A a1 ∈A1 ¯ (a1 , b1 )) ψ(f ψ(f (a, b)) b1 ∈B1 ,c1 ∈C1 b∈B,c∈C By using Lemma 2.1 and Cauchy’s inequality, it follows that Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only N− |A||B||C||A1 ||B1 ||C1 | q ≤ q ¯ ) (|B||C||B1 ||C1 |)1/2 q(D + (K − 1)q 1/2 ) ψ(a ψ(a) ψ∈Ψ∗ a∈A a1 ∈A1  ≤ (D + (K − 1)q 1/2 )(|B||C||B1 ||C1 |)1/2   1/2 ψ(a)  ψ∈Ψ a∈A 1/2 ¯  ψ(a) × ψ∈Ψ a∈A whence, by the identity zh ψ(h) ψ∈Ψ h∈Fq |zh |2 =q h∈Fq (for any complex number zh ∈ C), N− |A||B||C||A1 ||B1 ||C1 | q ≤ (D + (K − 1)q 1/2 )(|B||C||B1 ||C1 |)1/2 (q|A|)1/2 (q|A1 |)1/2 = q(D + (K − 1)q 1/2 )(|A||B||C||A1 ||B1 ||C1 |)1/2 , which completes the proof of Lemma 2.2 We are now ready to give a proof of Theorem 1.2 For any a ∈ Fq and three subsets A, B, C ⊆ Fq , denote by Na (A, B, C) the number of solutions of x1 + f (x2 , x3 ) = a with x1 ∈ A, x2 ∈ B, x3 ∈ C Note that |A + f (B, C)| = |{a : Na (A, B, C) > 0}|, (2.6) and Na (A, B, C) = |A||B||C| a∈Fq (2.7) March 1, 2014 9:43 WSPC/S1793-0421 696 203-IJNT 1350118 L A Vinh On the other hand, let T = |{(x1 , x2 , x3 , y1 , y2 , y3 ) ∈ A × B × C × A × B × C : x1 + f (x2 , x3 ) = y1 + f (y2 , y3 )}|, then Na2 (A, B, C) T = a∈Fq Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only It follows from Lemma 2.2 that T− |A|2 |B|2 |C|2 ≤ q(D + (K − 1)q 1/2 )|A||B||C| q This implies that T ≤ |A|2 |B|2 |C|2 + q(D + (K − 1)q 1/2 )|A||B||C| q (2.8) By Cauchy’s inequality, Na2 (A, B, C) ≥ |{a : Na (A, B, C) > 0}| a∈Fq Na (A, B, C) (2.9) a∈Fq Putting (2.6)–(2.9) together, we have |A + f (B, C)| ≥ |A|2 |B|2 |C|2 |A|2 |B|2 |C|2 + q(D + (K − 1)q 1/2 )|A||B||C| q q, |A||B||C| q(D + (K − 1)q 1/2 ) This concludes the proof of Theorem 1.1 The Expander x1 + x2 x3 We first take a detour to recall some graph theory concepts For a graph G with n vertices, 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 [1, Chap 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 March 1, 2014 9:43 WSPC/S1793-0421 203-IJNT 1350118 On Three-Variable Expanders Over Finite Fields 697 NU (v) = N (v) ∩ U and dU (v) = |NU (v)| We will need the following well-known fact (see, for example, [1]) Lemma 3.1 ([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 | ≤ λ |U ||W | n Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only 3.1 Sum–product graphs in finite fields Let Fq denote the finite field of q elements For any d ≥ 1, the sum–product graph SPq,d is defined as follows The vertex set of the sum–product graph SPq,d is the set V (SPq,d ) = Fq × Fdq Two vertices U = (a, b) and V = (c, d) ∈ V (SPq,d ) are connected by an edge, (U, V ) ∈ E(SPq,d ), if and only if a + c = b · d Our construction is similar to that of Solymosi in [27] Lemma 3.2 For any d ≥ 1, the sum–product graph, SPq,d , is a (q d+1 , q d , q d/2 )graph Proof It is easy to see that SPq,d is a regular graph of order q d+1 and of valency q d We now compute the eigenvalues of this multigraph For any a, c ∈ Fq and b = d ∈ Fdq , the system a + u = b · v, c + u = d · v, u ∈ Fq , v ∈ Fdq has q d−1 solutions (We can argue as follows There are q d−1 possibilities of v such that (b − d) · v = a − c For each choice of v, there exists a unique u satisfying the system.) If b = d and a = c, then the system has no solution Hence, for any two vertices U = (a, b) and V = (c, d) ∈ V (SPq,d ), if b = d then U and V have exactly q d−1 common neighbors, and if b = d and a = c then U and V have no common neighbors Let A be the adjacency matrix of SBq,d (λ) It follows that A2 = AAT = q d−1 J + (q d − q d−1 )I − q d−1 E, (3.1) where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph BE , where for any two vertices U = (a, b) and V = (c, d) ∈ V (SPq,d ), (U, V ) is an edge of BE if and only if a = c and b = d Since SPq,d is a q d -regular graph, q d is an eigenvalue of A with the all-one eigenvector The graph SPq,d is connected so the eigenvalue q d has multiplicity one Besides, choose b, d ∈ Fdq such that b · d = 2a = 0, then SPq,d contains a triangle with three vertices (−a, 0), (a, b), and (a, d), which implies that the graph is not bipartite Hence, for any other eigenvalue θ, |θ| < q d Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1⊥ , so Jv θ = It follows from (3.1) that (θ2 − q d + q d−1 )v θ = −q d−1 Ev θ (3.2) March 1, 2014 9:43 WSPC/S1793-0421 698 203-IJNT 1350118 L A Vinh Hence, v θ is also an eigenvector of E Since BE is a disjoint union of q d copies of the complete graph Kq , BE has eigenvalues q − with multiplicity q d , and −1 with multiplicity q d+1 − q d One corresponding eigenvector of the eigenvalue q − is the all-one eigenvector 1, and other corresponding eigenvectors are in the orthogonal space 1⊥ Plug into Eq (3.2), A has eigenvalues q d with multiplicity 1, and all other eigenvalues are ±q d/2 and The lemma follows Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only 3.2 Sum–product graphs in finite rings Let Zm = Z/mZm , be the finite cyclic rings of m elements Notice that we identify Zm with {0, 1, , m − 1} Similarly to the above, the sum–product graph SPm,d is defined as follows The vertex set of the sum–product graph SPm,d is the set V (SPm,d ) = Zm × Zdm Two vertices U = (a, b) and V = (c, d) ∈ V (SPm,d ) are connected by an edge, (U, V ) ∈ E(SPm,d ), if and only if a + c = b · d Theorem 3.3 For any d ≥ 1, the sum–product graph SPm,d is a md+1 , md , 2τ (m) md γ(m)d/2 − graph Proof It is easy to see that SPm,d is a regular graph of order md+1 and valency md We now compute the eigenvalues of this multigraph For any a, c ∈ Zm and b = d ∈ Zdm , we count the number of solutions of the following system: a+u =b·v c+u=d·v mod m, (b − d) · v = a − c mod m, mod m, u ∈ Zm , v ∈ Zdm (3.3) For each solution v of (3.4) there exists a unique u satisfying the system (3.3) Therefore, we only need to count the number of solutions of (3.4) Let n be the largest divisor of m such that all coordinates of b − d are also divisible by n If n a − c then (3.4) has no solution Suppose that n | a − c Let γ = (a − c)/n ∈ Zm/n and x = (b − d)/n ∈ Zdm/n We first count the number of solutions v ∈ Zdm/n of x·v = γ mod m/n (3.5) Suppose that m/n = pr11 prt t Let Si be the number of solutions v ∈ Zdpri of i x·v =γ mod pri i (3.6) Since n is the largest divisor of m such that all coordinates of b − d are also divisible by n, there exists an index xj which is not divisible by pi So for any choice of vk ∈ Zpri i , k = j, we can always find a unique vj ∈ Zpri i , which satisfies (3.6) r (d−1) This implies that Si = pi i By the Chinese Remainder Theorem, the number March 1, 2014 9:43 WSPC/S1793-0421 203-IJNT 1350118 On Three-Variable Expanders Over Finite Fields 699 of solutions of (3.5) is t t i=1 Int J Number Theory 2014.10:689-703 Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/10/15 For personal use only r (d−1) pi i Si = = (m/n)d−1 i=1 For each solution of (3.5), putting back into (3.4) gives us nd solutions Hence, (3.4) has md−1 n solutions if n | a − c Therefore, for any two vertices U = (a, b) and V = (c, d) ∈ V (SPm,d ), let n be the largest divisor of m such that all coordinates of b − d are also divisible by n, then U and V have md−1 n common neighbors if n | c − a and no common neighbors otherwise Let A be the adjacency matrix of SPm,d It follows that A2 = md−1 J + (md − md−1 )I − md−1 En n|m 1≤n