DSpace at VNU: The Number of Occurrences of a Fixed Spread amongn Directions in Vector Spaces over Finite Fields

Keywords Distinct angles· Finite Poincaré graphs · Projective rational geometry 1 Introduction LetFq denote the finite field with q elements where q 1 is an odd prime power.. The purpos

DOI 10.1007/s00373-012-1242-3

O R I G I NA L PA P E R

The Number of Occurrences of a Fixed Spread among n

Directions in Vector Spaces over Finite Fields

Le Anh Vinh

Received: 23 September 2008 / Revised: 20 September 2012

© Springer Japan 2012

Abstract We study a finite analog of a problem of Erd˝os, Hickerson and Pach on

the maximum number of occurrences of a fixed angle among n directions in

three-dimensional spaces

Keywords Distinct angles· Finite Poincaré graphs · Projective rational geometry

1 Introduction

LetFq denote the finite field with q elements where q  1 is an odd prime power

Here and throughout, the notation X  Y means that there exists C > 0 such that

X ≤ CY For any x, y ∈ F d , the distance between x , y is defined as x − y = (x1− y1)2+ + (x d − y d )2 Let E ⊂ Fd , d  2 Then the finite analog of the

classical Erd˝os distance problem is to determine the smallest possible cardinality of the set

viewed as a subset ofFq Bourgain et al [4], showed, using intricate incidence geom-etry, that for everyε > 0, there exists δ > 0, such that if E ∈ F2

q and C ε1q ε  |E| 

C ε2q2−ε, then|(E)|  Cδ|E|1+δ for some constants C ε1, C2

ε and C δ The relation-ship betweenε and δ in their argument is difficult to determine Going up to higher

dimension using arguments of Bourgain, Katz and Tao is quite subtle Iosevich and Rudnev [14] establish the following result using Fourier analytic method

L A Vinh (B)

University of Education, Vietnam National University, Hanoi, Vietnam

e-mail: vinhla@vnu.edu.vn

Theorem 1.1 ([14]) Let E ⊂ Fd such that |E|  Cq d /2 for C sufficiently large Then

|(E)|  min



q , |E|

q (d−1)/2



Iosevich and his collaborators investigated several related results using this method

in a series of papers [6,7,11–15] Using graph theoretic method, the author reproved some of these results in [18–20,22–24] The advantages of the graph theoretic method are twofold First, we can reprove and sometimes improve several known results in vector spaces over finite fields Second, our approach works transparently in the non-Euclidean setting In this note, we use the graph theoretic method to study a finite analog of a related problem of Erdos et al [10]

Problem 1.2 ([10]) Give a good asymptotic bounds for the maximum number of occurrences of a fixed angleγ among n unit vectors in three-dimensional spaces.

Ifγ = π/2, the maximum number of orthogonal pairs is known to be (n4/3 ) as

this problem is equivalent to bounding the number of point-line incidences in the plane (see [5] for a detailed discussion) For any other angleγ = π/2, we are far from good

estimates for the maximum number of occurrences ofγ The only known upper bound

is still O (n4/3 ), the same as for orthogonal pairs For the lower bound, Swanepoel and

Valtr [16] established the bound(n log n), improving an earlier result of Erdos et

al [10] It is, however, widely believed that the(n log n) lower bound can be much

improved

The purpose of this note is to study an analog of this problem in the three-dimension space over finite fields In vector spaces over finite fields, however, the separation of lines is not measured by the transcendental notion of angle A remarkable approach

of Wildberger [25,26] by recasting metrical geometry in a purely algebraic setting, eliminate the difficulties in defining an angle by using instead the notion of spread—in Euclidean geometry the square of the sine of the angle between two rays lying on those lines (the notation of spread will be defined precisely in Sect.2) Using this notation,

we now can state the main result of this note

Theorem 1.3 Let E be a set of unit vectors inF3

q with q3/2  |E|  q2 For any

γ ∈ F q , let f γ (E) denote the number of occurrences of a fixed spread γ among E.

Then f γ (E) = (|E|2/q) if 1 − γ is a square in F q and f γ (E) = 0 otherwise.

The rest of this note is organized as follows In Sect.2, we follow Wildberger’s construction of affine and projective rational trigonometry to define the notions of quadrance and spread We then define the main tool of our proof, the finite Poincaré graphs Using these graphs, we give a proof of Theorem1.3in Sect.3

2 Quadrance, Spread and Finite Poincaré Graphs

In this section, we follow Wildberger’s construction of affine and projective ratio-nal trigonometry over finite fields Interested readers can see [25,26] for a detailed discussion

2.1 Quadrance and Spread: Affine Rational Geometry

We work in a three-dimensional vector space over a field F , not of characteristic two.

Elements of the vector space are called points or vectors (these two terms are

equiv-alent and will be used interchangeably) and are denoted by U , V, W and so on The

zero vector or point is denote O The unique line l through distinct points U and V

is denoted U V For a non-zero point U the line OU is denoted [U] Fix a symmetric

bilinear form and represent it by U · V In terms of this form, the line U V is

perpen-dicular to the line W Z precisely when (V − U) · (Z − W) = 0 A point U is a null

point or null vector when U · U = 0 The origin O is always a null point, and there

may be others as well

The distance (or so-called quadrance in Wildberger’s construction) between the points U and V is the number

The line U V is a null line precisely when Q (U, V ) = 0, or equivalently when it is

perpendicular to itself

In Euclidean geometry, the separation of lines is traditionally measured by the

transcendental notion of angle The difficulties in defining an angle precisely, and in

extending the concept over an arbitrarily field, are eliminated in rational trigonometry

by using instead the notion of spread—in Euclidean geometry the square of the sine

of the angle between two rays lying on those lines Precisely, the spread between the non-null lines U W and V Z is the number

s (U W, V Z) = 1 − ((W − U) · (Z − V ))2

This depends only on the two lines, not the choice of points lying on them The spread between two non-null lines is 1 precisely when they are perpendicular Given a large

set E of unit vectors inF3, our aim is to study the number of occurrences of a fixed spreadγ ∈ F q among E.

2.2 Finite Poincaré Graphs: Projective Rational Geometry

Fix a three-dimensional vector space over a field with a symmetric bilinear form U · V

as in the previous subsection A line though the origin O will now be called a projec-tive point and denoted by a small letter such as u The space of such projecprojec-tive points

is called n dimensional projective space If V is a non-zero vector in the vector space,

thenv = [V ] denote the projective point OV A projective point is a null projective

point when some non-zero null point lies on it Two projective points u = [U] and

v = [V ] are perpendicular when they are perpendicular as lines.

The projective quadrance between the non-null projective points u = [U] and

v = [V ] is the number

q (u, v) = 1 − (U · U)(V · V ) (U · V )2 . (2.3)

This is the same as the spread s (OU, OV ), and has the value 1 precisely when the

projective points are perpendicular

The projective spread between the intersecting projective lines wu = [W, U] and

wv = [W, V ] is defined to be the spread between these intersecting planes:

S (wu, wv) = 1 −



U−U ·W

W ·W W



·V − V ·W

W ·W W

2



U− U ·W

W ·W W



·U− U ·W

W ·W W

 

V − V ·W

W ·W W



·V − V ·W

W ·W W



(2.4) This approach is entirely algebraic and elementary which allows one to formulate two dimensional hyperbolic geometry as a projective theory over a general field Pre-cisely, over the real numbers, the projective quadrance in the projective rational model

is the negative of the square of the hyperbolic sine of the hyperbolic distance between the corresponding points in the Poincaré model, and the projective spread is the square

of the sine of the angle between corresponding geodesics in the Poincaré model (see [26])

Let be the set of square-type non-isotropic 1-dimensional subspaces of F3then

|| = q(q + 1)/2 For a fixed γ ∈ F q, the finite Poincaré graph Pq (γ ) has vertices as

the points in and edges between vertices [Z], [W] if and only if s(O Z, OW) = γ

These graphs can be viewed as a companion of the well-known (and well studied) finite upper half plane graphs (see [17] for a survey on the finite upper half plane graphs)

From the definition of the spread, the finite Poincaré graph Pq (γ ) is nonempty if and

only if 1− λ is a square in F q.

The orthogonal group O3 (F q ) acts transitively on , and yields a symmetric

given by

R1= {([U], [V ]) ∈  ×  | (U + V ) · (U + V ) = 0},

R i = {([U], [V ]) ∈  ×  | (U + V ) · (U + V )

= 2 + 2ν −(i−1) } (2  i  (q − 1)/2)

R (q+1)/2 = {([U], [V ]) ∈  ×  · (U + V ) · (U + V ) = 2},

where ν is a generator of the field F q and we assume U · U = 1 for all [U] ∈ 

(see [2], Section 6) Note that 3(F q ), ) is isomorphic to the association scheme

P G L (2, q)/D2(q−1) where D2 (q−1) is a dihedral subgroup of order 2(q − 1) The

graphs(, R i ) are not Ramanujan in general, but fortunately, they are asymptotic

Ramanujan for large q The following theorem summaries the results from [3], Sect

2in a rough form

Theorem 2.1 ([3]) The graphs (, R i ) (1 ≤ i ≤ (q + 1)/2) are regular of valency

Cq (1 + o(1)) Let λ be any eigenvalue of the graph (, R i ) with λ = valency of the

graph then

|λ| ≤ c(1 + o(1))√q ,

for some C , c > 0 (In fact, we can show that c = 1/2).

Theorem2.1implies that the finite Poincaré graphs Pq (γ ) are asymptotic

Raman-ujan whenever 1− γ is a square in F q Precisely, we have the following theorem

Theorem 2.2 (a) If 1 − γ is not a square in F q then the finite Poincaré graph P q (γ )

is empty.

(b) If 1 −γ is a square in F q then the finite Poincaré graph P q (γ ) is regular of valency

Cq (1 + o(1)) Let λ be any eigenvalue of the graph P q (γ ) with λ = valency of

the graph then

|λ| ≤ c(1 + o(1))√q ,

for some C , c > 0.

Proof (a) Suppose that [U], [V ] ∈  and s(OU, OV ) = γ then

1− γ = (U · U)(V · V ) (U · V )2 .

But U , V are square-type so 1 − γ is a square in F q.

(b) It is easy to see that the finite Poincaré graphs Pq (1 − ν2−2i ) = (, R i ) for

1 ≤ i ≤ (q − 1)/2 and P q (1) = (, R (q+1)/2 ) The theorem follows

immedi-ately from Theorem2.1

3 Proof of Theorem 1.3

We call a graph G = (V, E) (n, d, λ)-regular if G is a d-regular graph on n vertices

with the absolute value of each of its eigenvalues but the largest one is at mostλ.

It is well-known that ifλ  d then a (n, d, λ)-regular graph behaves similarly as a

random graph Gn ,d/n Precisely, we have the following result (see Corollary 9.2.5 and Corollary 9.2.6 in [1])

Theorem 3.1 ([1]) Let G be a (n, d, λ)-regular graph For every set of vertices B of

G, we have



e(B) − 2n d  B|2|  1

where e (B) is number of edges in the induced subgraph of G on B.

Let E be a set of m unit vectors inF3then E can be viewed as a subset of  The

number of occurrences of a fixed spreadγ among E can be realized as the number of

edges in the induced subgraph of the finite Poincaré graph Pq (γ ) on the vertex set E.

Thus, from Theorem2.2, f γ (E) = 0 if 1 − γ is not a square in F q.

Suppose that 1− γ is a square in F q From Theorem2.2and Theorem3.1, we have

| fγ (E) − Cq (1 + o(1))

q (q + 1)/2 |E|2| ≤

1

2c (1 + o(1))√q |E|. (3.2) Since|E|  q3/2, we have1

2c (1+o(1))√q|E|  Cq (1+o(1))

q (q+1)/2 |E|2and the theorem follows

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