DSpace at VNU: On the generalized Erd˝ os–Falconer distance problems over finite fields

Journal of Number Theory 133 (2013) 2939–2947 Contents lists available at SciVerse ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt ˝ On the generalized Erdos–Falconer distance problems 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 October 2012 Revised October 2012 Accepted 10 October 2012 Available online May 2013 Communicated by David Goss Keywords: ˝ Erdos–Falconer distance problems Finite fields Pseudo-random graphs a b s t r a c t ˝ In this paper we study the generalized Erdos–Falconer distance problems in the finite field setting via spectra of directed graph The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained In particular, we give graph-theoretic proofs for various results due to Koh and Shen (2011) As a consequence, we also obtain a better bound under a weaker condition of the restricted sets © 2013 Elsevier Inc All rights reserved Introduction Let Fdq , d 2, be a d-dimensional vector space over the finite field Fq with q element, where q, a power of an odd prime, is viewed as an asymptotic parameter We shall work on the vector space Fdq , and throughout the paper, we shall assume that the characteristic of the finite field Fq is sufficiently large so that some minor technical problems can be overcome Here and throughout, the notation X Y means that there exists C > such that X C Y ˝ distance problem is to determine the For E ⊂ Fdq (d 2), the finite analogue of the classical Erdos smallest possible cardinality of the set (E ) = x − y = (x1 − y )2 + · · · + (xd − yd )2 : x, y ∈ E ⊂ Fq ˝ distance problem in vector spaces over finite fields is due to The first non-trivial result on the Erdos Bourgain, Katz, and Tao [2], who showed that if q is a prime, q ≡ (mod 4), then for every ε > E-mail address: vinhla@vnu.edu.vn This research was supported by Vietnam National Foundation for Science and Technology Development grant 101.012011.28 0022-314X/$ – see front matter © 2013 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.jnt.2012.10.015 2940 L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 and E ⊂ Fq2 with |E | q2− , there exists δ > such that | (E )| | E | +δ 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 [6] used Fourier analytic methods to show that for any odd prime power q and any set E ⊂ Fdq of cardinality |E | qd/2 , we have (E ) q, q d −1 |E | (1.1) Iosevich and Rudnev [6] reformulated the question in analogy with the Falconer distance problem: how large does E ⊂ Fdq , d 2, need to be to ensure that (E ) contains a positive proportion of d +1 the elements of Fq The above 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 (d + 1)/2 the distance set is of positive measure In [10], the author gave another proof of (1.1) using the graph-theoretic method (see also [13] for a similar proof) At first, it seemed reasonable that the exponent (d + 1)/2 may be improvable, in line with the Falconer distance conjecture described above However, Hart, Iosevich, Koh and Rudnev discovered in [5] that the arithmetic of the problem makes the exponent (d + 1)/2 best possible in odd dimensions, at least in general fields In even dimensions, it is still possible that the correct exponent is d/2, in analogy with the Euclidean case In [3], Chapman et al took a first step in this direction by showing that if E ⊂ Fq2 satisfies |E | q4/3 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 See also [3,11] for recent studies of distributions of simplexes in vector spaces over finite fields Following the similar techniques due to Iosevich and Rudnev [6], Koh and Shen [8] obtained the generalized distance formulas As an application of the formulas, they obtained results on the ˝ generalized Erdos–Falconer distance problems associated with specific diagonal polynomials P (x) = cj d a x In this paper, we give alternative proofs of these results via spectra of directed polyj =1 j j nomial graphs As a consequence, we also obtain a better bound under a weaker condition of the cj d restricted sets More precisely, let P (x) = cj N integers j =1 a j x j ∈ Fq [x1 , , xd ] for a j = 0, for some constant N > Let E , F ⊂ Fdq be product sets; i.e E = E × · · · × E d and F = F × · · · × F d If |E ||F | q2d /(2d−1) , then |{ P (x − y ) ∈ Fq : x ∈ E , y ∈ F }| q ˝ The generalized Erdos–Falconer distance problems We denote by χ : Fq → S1 the canonical additive character of Fq For example, if q is prime, then we can take χ (s) = e 2π is/q For the example of the canonical additive character of the general field Fq , see Chapter in [9] Let f : Fdq → C be a complex-valued function on Fdq Then, the Fourier transform of the function f is defined by f (m) = qd f (x)χ (−x · m) for m ∈ Fdq (2.1) x∈Fdq Let P (x) ∈ Fq [x1 , , xd ] be a polynomial with degree Given sets E , F ⊂ Fdq , recall that a generalized pair-wise distance set P (E , F ) is given by the set P (E , F ) = P (x − y ): x ∈ E , y ∈ F ⊂ Fq ˝ distance problems, one aims to find the lower bound of | P (E , F )| in terms of For the Erdos |E |, |F | For the Falconer distance problems, our goal is to determine an optimal exponent s0 > such that if |E ||F | q s0 , then | P (E , F )| q L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 2941 In this general setting, the main difficulty on these problems is that we not know the explicit form of the polynomial P (x) ∈ Fq [x1 , , xd ], generating generalized distances Koh and Shen found some conditions on the variety Vt = {x ∈ Fdq : P (x) = t } for t ∈ Fq such that some results can be obtained for the distance problems More precisely, they have the following distance formula Theorem 2.1 (See [8, Theorems 3.1 and 3.3].) Let E , F ⊂ Fdq and P (x) ∈ Fq [x1 , , xd ] For each t ∈ Fq , let Vt = x ∈ Fdq : P (x) − t = (2.2) Suppose that there is a set A ⊂ Fq such that |Vt | ∼ qd−1 for all t ∈ Fq \ A and V t (m) (a) If |E ||F | q− d +1 for all t ∈ / A , m ∈ Fdq \ (0, , 0) (2.3) qd+1 , then P (E , F ) q − | A | (b) Suppose further that | A | ∼ and Vt (m) If |E ||F | d q− for all t ∈ A , m ∈ Fdq \ (0, , 0) (2.4) qd , then P (E , F ) q, q− (d−1) |E ||F | The following lemma is well known and it was obtained by applying cohomological arguments d s Lemma 2.2 (See [4, Example 4.4.19].) Let P (x) = 2, a j = for all j = j =1 a j x j ∈ Fq [x1 , , xd ] with s 1, , d In addition, assume that the characteristic of Fq is sufficiently large so that it does not divide s Then, we have Vt (m) = qd χ (−x · m) q− d +1 for all m ∈ Fdq \ (0, , 0) , t ∈ Fq \ {0}, x∈Vt and V0 (m) d q− for all m ∈ Fdq \ (0, , 0) , where Vt = {x ∈ Fdq : P (x) = t } Combining Theorem 2.1 and Lemma 2.2 with the fact that |Vt | ∼ qd−1 for all t ∈ Fq \ {0}, Koh and Shen obtained a generalized version of spherical distance problems in [6] and cubic distance problems in [7] d s Theorem 2.3 (See [8, Corollaries 3.1 and 3.4].) Let P (x) = j =1 a j x j ∈ Fq [x1 , , xd ] for s a j = Suppose that the characteristic of Fq is sufficiently large integer and 2942 L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 qd+1 , then | (a) If E , F ⊂ Fdq of cardinality |E ||F | (b) If E , F ⊂ Fdq of cardinality |E ||F | P (E , F )| = q − d q , then q, q− P (E , F ) (d−1) |E ||F | In [12], the author gave graph-theoretic proofs of Theorem 2.1, Theorem 2.3 and related results More precisely, the author [12] showed that almost all systems of general equations are solvable in any large subset of vector spaces over finite fields We have seen that the distance problems are closely related to decays of the Fourier transforms on varieties In order to apply Theorem 2.1, we must estimate the Fourier decay of the variety Vt = {x ∈ Fdq : P (x) = t } In general, it is not easy to estimate the Fourier transform of Vt Koh and Shen [8] established a more useful, easier formula for distance problems than the formulas given in Theorem 2.1 In fact, they proved the following result Theorem 2.4 (See [8, Theorem 4.1].) Let P (x) ∈ Fq [x1 , , xd ] be a polynomial Given E , F ⊂ Fdq , define the distance set P (E , F ) = P (x − y ) ∈ Fq : x ∈ E , y ∈ F Suppose that the following estimate holds: for every m ∈ Fdq and s = 0, d χ s P ( x) + m · x q2 (2.5) x∈Fdq If |E ||F | qd+1 , then | P (E , F )| q Notice that the estimate (2.5) is easier than the estimate (2.3) Using Theorem 2.4, Koh and Shen [8] obtained the following result on the distance sets determined by the diagonal polynomials Theorem 2.5 (See [8, Corollary 4.2].) Let P (x) = gcd(c j , q) = for all j Let E , F ⊂ Fdq Define then | P (E , F )| q cj d j =1 a j x j ∈ Fq [x1 , , xd ] for c j integers, a j = 0, and P (E , F ) = { P (x − y ) ∈ Fq : x ∈ E , y ∈ F } If |E ||F | qd+1 Theorem 2.3 does not imply Theorem 2.5 above Considering the diagonal polynomial P (x) = if the exponents c j are distinct, then Theorem 2.3 does not give any information We cj d j =1 a j x j , have not found any reference which shows that for m ∈ Fdq \ {(0, , 0)}, and t = 0, Vt (m) d q− d +1 , cj where Vt = {x ∈ Fdq : j =1 a j x j = t } and all c j are not same Thus, we cannot apply Theorem 2.1 to obtain such result as in Theorem 2.5 In this paper, we extend our graph-theoretic method [12] to give another proof of Theorem 2.4 More precisely, we will prove the following variant of [8, Theorem 4.4] Theorem 2.6 Let P (x) ∈ Fq [x1 , , xd ] be a polynomial such that the estimate (2.5) holds Given A ⊂ Fq , E , F ⊂ Fdq , then A+ P (E , F ) q, | A ||E ||F | qd L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 2943 Notice that Theorem 2.4 follows immediately from Theorem 2.6 by letting A = {0} In addition, we also obtain a better bound for Theorem 2.5 under a weaker condition on the sets E and F Theorem 2.7 Let P (x) = cj d j =1 a j x j ∈ Fq [x1 , , xd ] for a j = 0, cj N integers for some constant N > Let E , F ⊂ Fdq be product sets; i.e E = E × · · · × E d and F = F × · · · × F d Define P (E , F ) If |E ||F | q2d /(2d−1) , then | P (E , F )| = P (x − y ) ∈ Fq : x ∈ E , y ∈ F q Pseudo-random digraphs Throughout this paper, let G be a directed graph (digraph) on n vertices where the in-degree and out-degree of each vertex are both d The adjacency matrix A G is defined as follows: j = if there is a directed edge from i to j and otherwise Let d = λ1 (G ), , λn (G ) be the eigenvalues of A G These numbers can be complex so we cannot order them, but by Frobenius’ theorem, all |λi | d Define λ(G ) := max |λi | i An n × n matrix A is normal if A t A = A A t , where A t is the transpose of A We say that a digraph is normal if its adjacency matrix is a normal matrix We call a digraph G an (n, d, λ)-digraph if G is normal and λ(G ) λ There is a simple way to test whether a digraph is normal In a digraph G, − → −→ let N + (x, y ) be the set of vertices z such that xz, yz are (directed) edges Similarly, let N − (x, y ) be − → − → the set of vertices z such that zx, zy are (directed) edges It is known that G is normal if and only if | N + (x, y )| = | N − (x, y )| for any two vertices x and y Let G be an (n, d, λ)-digraph 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 ∈ E (G ) (where E (G ) is the edge set of G) It is well known that if G is an (n, d, λ)-graph then G behaves like a random graph (see, for example, [1, Theorem 9.2.4]) Van Vu [13] developed a directed version of this result Lemma 3.1 (See [13, Lemma 3.1].) Let G = ( V , E ) be an (n, d, λ)-digraph For any two sets B , C ⊂ V and a color r, we have e( B , C ) − d n | B ||C | λ | B ||C | Let H be a finite (additive) abelian group and S be a subset of H Define a directed Cayley graph G S as follows The vertex of G is H There is a directed edge from x to y if and only if y − x ∈ S It is clear that every vertex G S has out-degree | S | Let χα , α ∈ H , be the additive characters of H It is well known that for any α ∈ H , s∈ S χα (s) is an eigenvalue of G S , with respect the eigenvector (χα (x))x∈ H It is easy to see that G S , for any S, is normal Given a polynomial Q (x) ∈ Fq [x1 , , xl ] and an element a ∈ Fq , now we define the graph G Q with the vertex set V = Fq × Flq and the edge set E (G Q ) = (x0 , x), ( y , y ) ∈ V × V (x0 , x) = ( y , y ), x0 − y + P (x − y ) = We have the following result of the spectra of the graph G Q and Q ∈ Fq [x1 , , xl ], the polynomial graph G Q is a Lemma 3.2 For any odd prime power q, l (ql+1 , ql , cql/2 )-digraph for some constant c > 2944 L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 Proof It is easy to see that G Q has ql+1 vertices and the degree of each vertex is ql Next, we will estimate eigenvalues of G Q Let V = {(x0 , x) ∈ Fq × Flq | x0 + P (x) = 0}, then the exponentials (or characters of the additive group Flq+1 ) χ(m0 ,m) (x0 , x) = χ (x0 , x) · (m0 , m) , (3.1) for (x0 , x), (m0 , m) ∈ Fq × Flq , are eigenfunctions of the adjacency operator for the graph G Q corresponding to the eigenvalue λ(m0 ,m) = χ(m0 ,m) (x0 , x) (x0 ,x)∈V χ (x0m0 + x · m) = x0 + P (x)=0 = = q χ s x0 + P (x) χ (x0m0 + x · m) (x0 ,x)∈Fq ×Fdq s∈Fq q χ s P (x) + x · m χ x0 (s + m0 ) s,x0 ∈Fq ,x∈Fdq χ −m0 P (x) + x · m , = x∈Fdq where the last line follows from the orthogonality of x0 The decay estimate (2.5) is equivalent to λ(m0 ,m) ql/2 , (3.2) for any (m0 , m) ∈ Fq∗ × Flq If (m0 , m) = (0, , 0), we have the corresponding eigenvalue λ(0, ,0) = ql If m0 = and m = (0, , 0) then λ(m0 ,m) vanishes This implies that G Q is a (ql+1 , ql , cql/2 )-digraph for some constant c > ✷ Proof of Theorem 2.6 We are now ready to give a graph-theoretic proof of Theorem 2.6 For any a ∈ Fq and A ⊆ Fq , E , F ⊆ Fdq , let N a ( A , E , F ) be the number of solutions of x1 + P (x2 − x3 ) = a with x1 ∈ A, x2 ∈ E , x3 ∈ F Note that A+ P (E , F ) = a: N a ( A , E , F ) > , (4.1) and N a ( A , E , F ) = | A ||E ||F | (4.2) a∈Fq On the other hand, let T= (x1 , x2 , x3 , y , y , y ) ∈ A × E × F × A × E × F : x1 + P (x2 − x3 ) = y + P ( y − y ) , L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 2945 then N a2 ( A , E , F ) T= a∈Fq Let U = A × E × E and W = A × F × F , then |U || W | = | A |2 |E |2 |F |2 and T becomes the number of edges between two vertex sets U , W of the polynomial graph G Q , where the polynomial Q ∈ Fq [x1 , , x2d ] is defined by Q (x1 , , x2d ) = P (x1 , , xd ) − P (xd+1 , , x2d ) It follows from Lemmas 3.1 and 3.2 that T− |U || W | qd |U || W | q This implies that T | A | | E | | F |2 q + qd | A ||E ||F | (4.3) By Cauchy’s inequality, N a2 ( A , E , F ) a: N a ( A , E , F ) > a∈Fq Na ( A , E , F ) (4.4) a∈Fq Putting (4.1), (4.2), (4.3), and (4.4) together, we have A+ P (E , F ) | A | | E | | F |2 | A |2 |E |2 |F |2 q + qd |A||E ||F | q, | A ||E ||F | qd This concludes the proof of Theorem 2.6 Proof of Theorem 2.7 Without loss of generality, we can assume that | E || F | = | E i || F i | i =1,d | E || F | 1/d Let d P (x2 , , xd−1 ) = cj a jxj , j =2 E = E2 × · · · × Ed, F = F2 × · · · × Fd, A = a1 (x1 − y )c1 x1 ∈ E , y ∈ F , (5.1) 2946 L.A Vinh / Journal of Number Theory 133 (2013) 2939–2947 then | A| | E − F |/c 1/2 | E || F | (5.2) We also need the following well-known Weil’s theorem We refer the readers to [9, Theorem 5.38] for a proof of Weil’s theorem Theorem 5.1 (Weil’s theorem) Let f ∈ Fq [x] be the polynomial of degree c have with gcd(c , q) = Then, we (c − 1)q1/2 , χ f (s) x∈Fq where χ denotes a non-trivial additive character of Fq It follows immediately from Theorem 5.1 that χ s P ( x) + m · x q d −1 (5.3) x∈Fdq−1 for every m ∈ Fdq−1 and s = From (5.3) and Theorem 2.6, we have A+ P E ,F q, | A ||E ||F | q d −1 (5.4) Putting (5.1), (5.2), and (5.4) together, we have P (E , F ) = A+ P q, q, q, E ,F | A ||E ||F | q d −1 | E || F | 1/2 || F |) qd−1 (| E (| E || F |) 2d−1 2d q d −1 q, concluding the proof of Theorem 2.7 References [1] N Alon, J.H Spencer, The Probabilistic Method, 2nd edition, Wiley–Interscience, 2000 [2] J Bourgain, N Katz, T Tao, A sum product estimate in finite fields and applications, Geom Funct Anal 14 (2004) 27–57 [3] J Chapman, M.B Erdogan, Derrick Hart, Alex Iosevich, Doowon Koh, Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates, Math Z 271 (1–2) (2012) 63–93 [4] T Cochrane, Exponential Sums and the Distribution of Solutions of Congruences, Inst of Math., Academia Sinica, Taipei, 1994 [5] D Hart, A Iosevich, D Koh, M Rudnev, Averages over hyperplanes, 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(published online) [12] L.A Vinh, Solvability of systems of general equations over finite fields, Indiana Univ Math J., in press [13] V.H Vu, Sum-product estimates via directed expanders, Math Res Lett 15 (2) (2008) 375–388  ... setting, the main difficulty on these problems is that we not know the explicit form of the polynomial P (x) ∈ Fq [x1 , , xd ], generating generalized distances Koh and Shen found some conditions on. .. direction by showing that if E ⊂ Fq2 satisfies |E | q4/3 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... Following the similar techniques due to Iosevich and Rudnev [6], Koh and Shen [8] obtained the generalized distance formulas As an application of the formulas, they obtained results on the ˝ generalized