2 Association scheme definitions

See [Reference Brouwer and Haemers5] for a discussion of the essentials of the theory of association schemes. There are various equivalent definitions of association schemes, for us the following is most convenient.

The best way to think of this last condition is to think of $x,y$ giving a segment of length k and then noting that (3) states that the number of ways to complete this segment into a “triangle” only depends on the side lengths of the triangle and not the endpoints of the segment itself.

Though it is nice to think of an association scheme “geometrically” do note that we do not require the “distance” to satisfy the triangle inequality nor for the set $\Delta $ to be numerical. The elements of $\Delta $ can be any objects such as colors, etc.

For any $x \in X$ and $j \in \Delta $ , the sphere of radius j about x is denoted

$$ \begin{align*}S_j(x)=\{ y \in X | d(x,y)=j \}.\end{align*} $$

Note in any finite association scheme, setting $x=y$ in $(3),$ we see that

$$ \begin{align*}p_{i,j}^{\bar{0}}=\delta_{i,j} |S_j(x)|\end{align*} $$

is independent of x. Here, $\delta _{i,j}$ is the Kronecker Delta function which is $1$ when $i=j$ and $0$ otherwise. Furthermore, by (2), we may reverse the role of x and y and so conclude that

$$ \begin{align*}p_{i,j}^k = p_{j,i}^k\end{align*} $$

for all $i,j,k \in \Delta $ .

Given an association scheme $(X,d)$ , we define the distance-t graph for any $t \in \Delta $ as the graph on vertex set X and where two vertices $x, y \in X$ are adjacent if and only if $d(x,y)=t$ . As $|S_t(x)|$ is independent of x, the distance-t graph is $|S_t|$ -regular.

In the case X is finite, we denote the adjacency matrix of the graph (with respect to some linear ordering of X) as $\mathbb {A}_t$ . Thus, $\mathbb {A}_{\bar {0}} = I$ and $\sum _{j \in \Delta } \mathbb {A}_j = \mathbb {J}$ , where $\mathbb {J}$ is the all $1$ matrix. It is easy to check that condition (2) is equivalent to the $\mathbb {A}_t$ being symmetric matrices and condition (3) is equivalent to

$$ \begin{align*}\mathbb{A}_i \mathbb{A}_j = \sum_{k} p_{i,j}^k \mathbb{A}_k. \end{align*} $$

As $p_{i,j}^k=p_{j,i}^k$ , this says the matrices $\{ \mathbb {A}_i | i \in \Delta \}$ all commute with each other. Note as we require $d: X \times X \to \Delta $ to be surjective, no $\mathbb {A}_j$ is the zero matrix and so the fact that they sum to $\mathbb {J}$ shows that the $\{ \mathbb {A}_i | i \in \Delta \}$ are a linearly independent set in $Mat_{|X|}(\mathbb {R})$ . Condition (3), guarantees that their span is an algebra.

Definition 2.2 (Bose–Messner algebra)

Let $\mathfrak {A}=(X,d)$ be a finite association scheme with distance set $\Delta $ such that $|\Delta |=D+1$ . We say $(X,d)$ has D (nonzero) classes. The Bose–Messner algebra of the scheme is defined as

$$ \begin{align*}BM(\mathfrak{A})=\mathbb{R}-\text{span of } \{ \mathbb{A}_j | j \in \Delta \} \subseteq Mat_{|X|}(\mathbb{R}).\end{align*} $$

It is a ( $D+1$ )-dimensional $\mathbb {R}$ -commutative algebra with basis $\{ \mathbb {A}_j | j \in \Delta \}$ and multiplication determined by the rule $\mathbb {A}_i \mathbb {A}_j = \sum _{k} p_{i,j}^k \mathbb {A}_k$ .

The complexification of this algebra is called the complex Bose–Messner algebra and consists of the $\mathbb {C}$ -span of the same $\mathbb {A}_j$ matrices.

The next example is the primary example considered in this paper.

Example 2.4 (Euclidean association scheme)

Let q be an odd prime power, $n\geq 2$ , and $V=\mathbb {F}_q^n$ equipped with the “Euclidean” quadratic form $Q(v)=v_1^2+\dots +v_n^2$ and “distance” $Q(v-w)$ . Then given $x \neq y,v \neq w \in V$ with $Q(x-y)=Q(v-w)$ , Witt’s theorem shows there is a isometry of V (consisting of a composition of a translation and matrix multiplication by A, where $A \in O(n,q)=\{ \mathbb {A} \in Mat_n(\mathbb {F}_q) | A^T A = I \}$ is an “orthogonal” matrix), taking the pair $x,y$ to the pair of $v,w$ .

When $n \geq 3$ or $n=2$ and $q=1 \text { mod } 4$ , it is possible $Q(x-y)=0$ but $x \neq y$ and so we distinguish between distance $0$ and distance $\bar {0}$ by declaring $d(x,y)=\bar {0}$ if and only if $x=y$ and $d(x,y)=Q(x-y)$ whenever $x \neq y$ . Thus in $\mathbb {Z}_5^2$ ,

$$ \begin{align*}d((0,0),(1,2))=1^2+2^2=5=0 \neq \bar{0}.\end{align*} $$

The distance set is $\Delta =\mathbb {F}_q \cup \{ \bar {0} \}$ when $n \geq 3$ or $n=2, q=1 \text { mod } 4$ but when $n=2, q=3 \text { mod } 4$ , $\Delta =(\mathbb {F}_q-\{ 0\})\cup \{ \bar {0} \}$ as $Q(x-y) \neq 0$ when $x \neq y$ in that case.

Witt’s theorem establishes that $(V,d)$ is an association scheme by establishing property (3).

Example 2.5 Let q be an odd prime power, $q = 3 \text { mod 4}$ , then $-1$ is not a square in $\mathbb {F}_q$ and so the Galois extension $E=\mathbb {F}_q[i]$ of $\mathbb {F}_q$ is a degree $2$ extension where i is a root of $x^2+1$ . The Galois norm map $N: E^{\times } \to \mathbb {F}_q^{\times }$ has $N(a+bi)=(a+bi)(a-bi)=a^2+b^2$ and this agrees with the Euclidean quadratic form Q on $V=\mathbb {F}_q^2$ and so $Q(x-y)=0 \iff x=y$ in this case. Note as N is a homomorphism,

$$ \begin{align*}|S_j((0,0))|=|N^{-1}(j)|=\frac{q^2-1}{q-1}=q+1\end{align*} $$

for all $j \in \mathbb {F}_q^{\times } = \mathbb {F}_q-\{ 0 \}.$ Thus, in the corresponding association scheme, $p_{j,j}^{\bar {0}} = q+1$ whenever $j \in \mathbb {F}_q^{\times }$ while $p_{\bar {0},\bar {0}}^{\bar {0}}=1$ .

Given a finite association scheme $\mathfrak {A}=(X,d)$ on D classes, the matrices $\{ \mathbb {A}_j | j \in \Delta \}$ are a real basis for the corresponding Bose–Messner algebra $BM(\mathfrak {A})$ . As the matrices are symmetric and as they commute, they can be simultaneously (orthogonally) diagonalized on their action on $\mathbb {R}^X$ and this vector space splits as an orthogonal direct sum of simultaneous eigenspaces of all the $\mathbb {A}_j$ operators. These simultaneous eigenspaces are called weight spaces. Let $\{ E_i, i \in \Delta ' \}$ be the collection of orthogonal projection operators to the individual weight spaces. Linear algebra guarantees that each $E_i$ is a polynomial expression in the $A_j$ ’s and hence lies in the Bose–Messner algebra and thus is a linear combination of the $A_j$ . Conversely, as each $A_j$ is constant on the image of any $E_i$ , each $A_j$ can be trivially written as a linear combination of the $E_i$ . Thus, the $\{ E_i | i \in \Delta ' \}$ are a different, “spectral basis” of the Bose–Messner algebra $BM(\mathfrak {A})$ and hence $|\Delta | = |\Delta '|$ . We will refer to the original basis $\{ \mathbb {A}_j | j \in \Delta \}$ of $BM(\mathfrak {A})$ as the geometric basis of the Bose–Messner algebra as these encode the distances of the scheme.

Many of the main results of association schemes, and distance regular graphs, in particular, arise from the interactions between the geometric and spectral basis of the Bose–Messner algebra of the scheme.

In particular, two fundamental change of basis matrices are defined as follows.

Definition 2.6 Let $(X,d)$ be a finite association scheme on D (nonzero) classes, and let $\{ A_j | j \in \Delta \}$ and $\{ E_i | i \in \Delta '\}$ denote the “geometric” and “spectral” basis of the corresponding Bose–Messner algebra. The $(D+1) \times (D+1)$ real matrices P and Q are defined via

$$ \begin{align*}A_j = \sum_{i \in \Delta'} P_{i,j} E_i \text{ for all } j \in \Delta \end{align*} $$

and

$$ \begin{align*}E_i = \frac{1}{|X|} \sum_{j \in \Delta} Q_{j,i} A_j \text{ for all } i \in \Delta'. \end{align*} $$

Thus $P Q = |X| I = Q P$ .

Note that the $\{ P_{i,j} | i \in \Delta '\}$ are just the eigenvalues of $A_j$ , the adjacency matrix of the distance-j graph of the scheme. Note that $|\Delta '|=|\Delta |=D+1$ is usually much smaller than $|X|$ and so the $|X| \times |X|$ matrix $A_j$ has lots of multiplicities in its eigenvalues, encoded by $rank(E_i)$ which is the dimension of the ith weight space.

The matrices $P, Q, L_j, A_i, E_j$ satisfy many interesting identities in an association scheme. For our purpose, we focus on Delsarte’s linear programming bound. Let $(X,d)$ be a finite association scheme and $Y \subseteq X$ . For each $j \in \Delta $ , define $a_j=\frac {| \{ (y_1,y_2) \in Y \times Y | d(y_1,y_2)=j \}|}{|Y|} =\frac {\chi _Y^T A_j \chi _Y}{\chi _Y^ T \chi _Y} \in \mathbb {Q}_{\geq 0}$ , where $\chi _Y$ is the characteristic column vector of Y. Then, the row vector $a=(a_j)_{j \in \Delta }$ is called the inner distribution of Y as it encodes the frequency of various distances on the pairs in the subset Y. This inner distribution satisfies Delsarte’s linear programming bound condition $a Q \geq 0$ as well as $\sum _{j \in \Delta } a_j = |Y|$ and these conditions give us many constraints on distances induced on subsets Y of the scheme.

3 Eigenvalues of the distance-t graphs

For any $t \in \mathbb {F}_q$ , let $X(q,t)$ denote the associated distance-t graph on $\mathbb {F}_q^d$ . In this section, we describe the spectrum of the adjacency matrix of this graph, $A_t$ , when $t \neq \bar {0}$ , i.e., for $t \in \mathbb {F}_q$ . This spectrum was previously computed by various authors, for example, in [Reference Iosevich and Rudnev9, Reference Terras17] but we include a brief discussion here to be self-contained.

Standard spectral theory of finite Cayley graphs over Abelian groups, shows that the complex eigenfunctions of this graph’s adjacency operator $A_t$ are the characters $\chi _m(x)=\chi (m \cdot x)$ as m ranges over $\mathbb {F}_q^d$ . Here, $\cdot $ stands for dot product. The corresponding eigenvalue of $A_t$ on $\chi _m$ which we will denote $\lambda _{m,t,d}$ (recall d is the dimension of V) is given by

$$ \begin{align*}\lambda_{m,t,d} =\sum_{x \in S_{t,d}} \chi_m(x) = \sum_{x \neq 0, Q(x)=t} \chi(m \cdot x). \end{align*} $$

Clearly, $\lambda _{0,t,d} = |S_{t,d}|$ . The size of these spheres were originally calculated by Minkowski, and there are at most three distinct sphere sizes corresponding to $t=0$ , t a nonsquare in $\mathbb {F}_q$ , and t a nonzero square in $\mathbb {F}_q$ . This is because scaling by $\lambda \in \mathbb {F}_q^{\times }$ gives a bijection between $S_{t,d}$ and $S_{\lambda ^2 t, d}$ .

When $m,t$ are both nonzero, these eigenvalues are given by twisted Kloosterman sums. We provide the calculation for the convenience of the reader. First, note that by character orthogonality, we have $\frac {1}{q}\sum _{s \in \mathbb {F}_q} \chi (sL)=\delta _{L,0}$ , where $\delta _{L,0}=1$ if $L=0$ and $0$ otherwise. Then we compute, when $t \neq 0$ ,

$$ \begin{align*} \lambda_{m,t,d} &= \sum_{\{x | Q(x)=t\}} \chi(m \cdot x) =\frac{1}{q} \sum_{s \in \mathbb{F}_q} \sum_{x \in V} \chi(m \cdot x) \chi(s(Q(x)-t)) \\ &= \frac{1}{q}\sum_{s \in \mathbb{F}_q} \chi(-st) \sum_{x_1,\dots, x_d \in \mathbb{F}_q} \chi(m_1x_1 + \dots + m_dx_d + s(x_1^2 + \dots + x_d^2)) \\ &=\frac{1}{q}\sum_{s \in \mathbb{F}_q} \chi(-st) \prod_{i=1}^d \sum_{x_i \in \mathbb{F}_q} \chi(m_i x_i + sx_i^2) \\ &=q^{d-1} \delta_{m,0} + \frac{1}{q}\sum_{s \neq 0} \chi(-st) \prod_{i=1}^d \sum_{x_i \in \mathbb{F}_q} \chi(m_i x_i + sx_i^2) \\ &=q^{d-1} \delta_{m,0} + \frac{1}{q}\sum_{s \neq 0} \chi(-st) \prod_{i=1}^d \sum_{y_i \in \mathbb{F}_q} \chi(sy_i^2-\frac{m_i^2}{4s}), \end{align*} $$

where in the last step, we completed the square and substituted $y_i=x_i+\frac {m_i}{2s}$ . Letting $G(s)=\sum _{y \in \mathbb {F}_q} \chi (sy^2)$ denote the corresponding Gauss sum, we conclude that

$$ \begin{align*}\lambda_{m,t,d}=q^{d-1} \delta_{m,0} + \frac{1}{q}\sum_{s \neq 0} \chi\left(-st-\frac{Q(m)}{4s}\right) G(s)^d. \end{align*} $$

When $t=0,$ the only correction needed is to exclude $x=0$ from the set $Q(x)=0$ which results in reducing this quantity by $1$ . Thus, we have generally that

$$ \begin{align*}\lambda_{m,t,d}=q^{d-1} \delta_{m,0} -\delta_{t,0} + \frac{1}{q}\sum_{s \neq 0} \chi\left(-st-\frac{Q(m)}{4s}\right) G(s)^d. \end{align*} $$

It is a well-known result of Gauss that $G(s)=\binom {s}{q} \epsilon (q) \sqrt {q}$ , where $\binom {s}{q}$ is the Legendre symbol which is $+1$ when s is a nonzero square modulo q, and $-1$ when s is a nonsquare modulo q. The quantity $\epsilon (q)=1$ when $q \equiv 1 \text { mod } 4$ and $\epsilon (q)=i$ when $q \equiv 3 \text { mod } 4$ . Thus, we conclude

$$ \begin{align*}\lambda_{m,t,d}=q^{d-1} \delta_{m,0} -\delta_{t,0} + q^{\frac{d}{2}-1} \epsilon(q)^d \tilde{K}_d\left(-t,-\frac{Q(m)}{4}\right), \end{align*} $$

where $\tilde {K}_d(a,b)=\sum _{s \neq 0} \binom {s}{q}^d \chi (as+\frac {b}{s})$ is a twisted Kloosterman sum. Note that the dependence of $\tilde {K}_d(a,b)$ on d is weak and only depends on the parity of d. When d is even, $\tilde {K}_d(a,b)=K(a,b)=\sum _{s \neq 0} \chi (as+\frac {b}{s})$ is the regular Kloosterman sum. Note that the m-dependence of the eigenvalue $\lambda _{m,t,d}$ is only through $Q(m)=||m||=m_1^2+\dots +m_d^2$ and whether m is the origin or not. Following the convention adopted in association schemes, we will write $Q(m)=\bar {0}$ if $m=0$ and $Q(m)=0$ if $m \neq 0$ but $m_1^2+\dots +m_d^2=0$ .

Thus $Span\{\chi _m | Q(m)=k\}=L_k$ is contained in an eigenspace of $A_t$ for each $t,k \in \mathbb {F}_q \cup \{ \bar {0} \}$ . The complex vector space $\mathbb {C}[V]$ , of all complex functions on V, decomposes as a orthogonal direct sum of the $L_k$ with respect to the standard Hermitian inner product. The $L_k$ are simultaneous eigenspaces of all the $A_t$ operators and $L_0$ is $0$ if and only if $S_0 = \emptyset $ . Thus, the number of nonzero $L_k$ is the same as the dimension of the corresponding Euclidean association scheme and so the nonzero $L_k$ are exactly the weight spaces in the decomposition of the complex Bose–Messner algebra of the scheme. Note that $L_{\bar {0}}=Span\{\chi _0\} =Span\{ \mathfrak {1} \}$ consists of the constant functions.

We record these results in the following theorem.

Theorem 3.1 Let q be an odd prime power and $\mathbb {F}_q$ the finite field of order q. For d an integer $\geq 2$ , $m \in \mathbb {F}_q^d, t \in \mathbb {F}_q$ we have that the eigenvalue $\lambda _{m,t,d}$ of the adjacency matrix $A_t$ of the distance-t graph of $V=\mathbb {F}_q^d$ on the character $\chi _m(x)=\chi (m \cdot x)$ is given by

$$ \begin{align*} \lambda_{m,t,d}=q^{d-1} \delta_{m,0} -\delta_{t,0} + q^{\frac{d}{2}-1} \epsilon(q)^d \tilde{K}_d\left(-t,-\frac{Q(m)}{4}\right), \end{align*} $$

where $\tilde {K}_d(a,b)=\sum _{s \neq 0} \binom {s}{q}^d \chi (as+\frac {b}{s})$ is a twisted Kloosterman sum.

If $W_k=Span\{\chi _m | Q(m)=k\}$ in $\mathbb {C}[V]$ , then $\mathbb {C}[V]=\oplus _{k \in \mathbb {F}_q \cup \{ \bar {0} \}} W_k$ is an orthogonal direct sum decomposition of the vector space of complex-valued functions on V into simultaneous eigenspaces of all the $A_t$ operators, $t \in \mathbb {F}_q \cup \{ \bar {0}\}$ . The orthogonal projection $E_k$ onto the (nonzero) $W_k$ give the spectral basis of the complex Bose–Messner algebra of the corresponding Euclidean association scheme.

Note it is not hard to see that the complex conjugate of $\tilde {K}_d(a,b)$ is

$$ \begin{align*} \sum_{s \neq 0} \binom{s}{q}^d \chi\left(-as - \frac{b}{s}\right)=\sum_{s \neq 0} \binom{-s}{q}^d \chi\left(as + \frac{b}{s}\right)\end{align*} $$

and so $\tilde {K}_d(a,b)$ is real if either d is even or $q\equiv 1 \text { mod } 4$ (i.e., when $-1$ is a square modulo q) but is purely imaginary when d is odd and $q \equiv 3 \text { mod 4}$ . However, the quantity $\epsilon (q)^d \tilde {K}_d(a,b)$ and hence $\lambda _{m,t,d}$ are real numbers of course as they are eigenvalues of symmetric matrices.

4 Some properties of (twisted) Kloosterman sums

Let $q=p^{\ell }$ be an odd prime power and define $\tilde {K}_d(a,b)=\sum _{s \neq 0} \binom {s}{q}^d \chi (as+\frac {b}{s})$ for $a,b \in \mathbb {F}_q$ .

Let $\xi _p=e^{\frac {2 \pi i}{p}}$ be a primitive pth root of unity and $\mathbb {Q}[\xi _p]$ be the corresponding cyclotomic field. Note as $\chi (x)=\xi ^{\mathrm{Tr}(x)}$ , where $\mathrm{Tr}: \mathbb {F}_q \to \mathbb {F}_p$ is the Galois trace, we have that $\tilde {K}_d(a,b)$ are elements in the ring of integers of $\mathbb {Q}[\xi _p]$ . Recall that the Galois group $Gal(\mathbb {Q}[\xi _p]/\mathbb {Q})$ is Abelian of order $p-1$ and contains automorphisms of the form $\psi _a$ determined by the property that they take $\xi $ to $\xi ^a$ for all $a \in \mathbb {F}_p^*$ and fix the rational subfield.

Proposition 4.1 Let p be an odd prime and $q=p^{\ell }$ and define

$$ \begin{align*}\tilde{K}_d(a,b)=\sum_{s \neq 0} \binom{s}{q}^d \chi\left(as+\frac{b}{s}\right)\end{align*} $$

for $a,b \in \mathbb {F}_q$ . These are algebraic integers in the cyclotomic field $\mathbb {Q}[\xi _p]$ which satisfy:

(1) $\tilde {K}_d(a,b)=\tilde {K}_d(b,a).$

(2) $\tilde {K}_d(a,b)=\binom {a}{q}^d\tilde {K}_d(1,ab)$ when $a \neq 0.$

(3) The quantity $\tilde {K}_d(0,0)$ is equal to $q-1$ when d is even and equal to $0$ when d is odd. Thus $\tilde {K}_d(0,0)=(q-1)\frac {(-1)^d+1}{2}$ .

(4) The quantity $\tilde {K}_d(1,0)=\tilde {K}_d(0,1)$ equals $-1$ when d is even and $G(1)$ when d is odd where $G(1)=\sum _{s \in \mathbb {F}_q} \chi (s^2)=\epsilon (q)\sqrt {q}$ is a quadratic Gauss sum.

(5) For $c \in \mathbb {F}_p^*$ , we have that $\psi _c(\tilde {K}_d(a,b))=\tilde {K}_d(ca,cb),$ where $\psi _c$ is the Galois automorphism of $\mathbb {Q}[\xi _p]$ over $\mathbb {Q}$ which takes $\xi _p$ to $\xi _p^c$ .

Proof To prove (1), use the change of variable $s \to \frac {1}{s}$ . To prove (2), use the change of variable $s'=as$ for nonzero a. (3) follows trivially once one notes that in any finite field, an equal number of elements are nonzero squares as are nonsquares.

For (4), first note that when d is even we have $\tilde {K}_d(1,0)=\sum _{s \neq 0} \chi (s)$ which by character orthogonality is equal to $-\chi (0)=-1$ . When d is odd, we have $\tilde {K}_d(1,0)=\sum _{s \neq 0} \binom {s}{q}\chi (s)=\sum _{s \in \mathbb {F}_q} \binom {s}{q}\chi (s)=\sum _{s \in \mathbb {F}_q} (\binom {s}{q}+1)\chi (s)=\sum _{u \in \mathbb {F}_q} \chi (u^2)=G(1).$

Finally, for (5), we note that $\tilde {K}_d(a,b)=\sum _{s \neq 0} \binom {s}{q}^d \xi _p^{\mathrm{Tr}(as+\frac {b}{s})}$ and apply the Galois automorphism $\psi _c$ to get

$$ \begin{align*}\psi_c(\tilde{K}_d(a,b)) = \sum_{s \neq 0} \binom{s}{q}^d \xi_p^{c\mathrm{Tr}(as+\frac{b}{s})}=\sum_{s \neq 0} \binom{s}{q}^d \xi_p^{\mathrm{Tr}(cas+\frac{cb}{s})}=\tilde{K}_d(ca,cb)\end{align*} $$

as the Galois trace $\mathrm{Tr}: \mathbb {F}_q \to \mathbb {F}_p$ is $\mathbb {F}_p$ -linear.

Note that when d is even, the $\tilde {K}_d(a,b)$ coincide with (untwisted) Kloosterman sums $K(a,b)=\sum _{s \neq 0} \chi (as+\frac {b}{s})$ and the properties above apply to those also. In particular, when $q=p$ , this implies that the $\{ K(1,u) | u \text { a nonzero square} \}$ are all Galois conjugates in $\mathbb {Q}[\xi _p]$ . This is because for $c \neq 0$ , $\psi _c(K(1,1))=K(c,c)=K(1,c^2)$ .

We will need one more property of these Kloosterman numbers that requires some Fourier analysis. Recall if $f: V \to \mathbb {C}$ , then we define the Fourier transform

$$ \begin{align*}\hat{f}(m)=\frac{1}{q^d}\sum_{x \in V} f(x) \chi(-m \cdot x).\end{align*} $$

It is easy to show via character orthogonality that we can recover f via

$$ \begin{align*}f(x) = \sum_{m \in V} \hat{f}(m)\chi(m \cdot x).\end{align*} $$

Note if we let $S_t$ denote the indicator function of the corresponding t-sphere, $t \in \Delta $ , then $q^d \hat {S}_t(m)=\sum _{x \in \mathbb {F}_q^d} S_t(x)\chi (-m \cdot x) = \sum _{x \in S_t} \chi (m \cdot x) = \lambda _{m,t,d}$ from the previous section. (We define $\lambda _{m,\bar {0},d}=1$ to also make it work in that case.) By disjointness of spheres of different radii, it follows that for any $i \neq j \in \Delta $ ,

$$ \begin{align*} 0 = S_i(x)S_j(x) &=\left(\sum_{m \in V} \hat{S}_i(m)\chi(m \cdot x) \right)\left(\sum_{n \in V} \hat{S}_j(n) \chi(n \cdot x)\right) \\ 0 &=\sum_{t \in V} \left(\sum_{m \in V} \hat{S}_i(m)\hat{S}_j(t-m)\right) \chi(t \cdot x). \end{align*} $$

By character independence, we conclude that $\sum _{m \in V} \hat {S}_i(m) \hat {S}_j(t-m) = 0$ for all $t \in \mathbb {F}_q^d$ when $i \neq j \in \Delta $ . When $i=j$ , as $S_i(x)S_i(x)=S_i(x)$ , we instead conclude $\hat {S}_i(t) = \sum _{m \in V} \hat {S}_i(m) \hat {S}_i(t-m)$ .

Using that $q^d \hat {S}_t(m)=\lambda _{m,t,d}$ and that $\hat {S}_t(m)=\hat {S}_t(-m)$ , this becomes the next proposition.

Proposition 4.2 Consider the Euclidean association scheme on $V=\mathbb {F}_q^d$ with distance set $\Delta $ . Let $\lambda _{m,t,d}$ be as in Theorem 3.1 with $\lambda _{m,\bar {0},d}=1$ . Then for any $i \neq j \in \Delta $ and $t \in V$ , we have

$$ \begin{align*}\sum_{m \in V} \lambda_{m,i,d} \lambda_{t-m,j,d} = 0, \end{align*} $$

and, in particular, when $t=0$ ,

$$ \begin{align*}\sum_{m \in V} \lambda_{m,i,d} \lambda_{m,j,d} = 0. \end{align*} $$

When $i=j$ , we instead have

$$ \begin{align*}\sum_{m \in V} \lambda_{m,i,d} \lambda_{t-m,i,d} = q^d \lambda_{t,i,d} \end{align*} $$

and

$$ \begin{align*}\sum_{m \in V} \lambda_{m,i,d}^2 = q^d \lambda_{0,i,d}=q^{2d}\hat{S}_i(0)=q^d|S_i|. \end{align*} $$

5 The P and Q matrices of the Euclidean association scheme

The calculation of the P-matrix of the Euclidean scheme in this section was first done in [Reference Kwok12] with corrections in [Reference Bannai, Shimabukuro and Tanaka2] but we provide the details independently here in language more connected to our applications.

Let $\Delta $ be the “distance” set of the Euclidean association scheme on $V=\mathbb {F}_q^d$ . Thus, if $d \geq 3$ or $(d=2, q=1 \text { mod } 4)$ , we have $\Delta =\{ \bar {0} \} \cup \mathbb {F}_q$ whereas when $(d=2, q=3 \text {mod } 4)$ , we have $\Delta =\{ \bar {0} \} \cup \mathbb {F}_q^{\times }$ .

By Theorem 3.1, it follows that both the spectral and geometric basis of the complex Bose–Messner algebra can be indexed by the distance set $\Delta $ . Fix a linear ordering of $\Delta $ where $\bar {0}$ comes first and if $0 \in \Delta $ , it comes second. For example, the ordering $\bar {0} < 0 < 1 < \dots < q-1$ works when $(d=2, q=1 \text { mod } 4)$ or $d \geq 3$ and when otherwise, i.e., when $(d=2, q=3 \text { mod } 4)$ we drop $0$ from the list.

Then when $i, j \in \Delta $ , we have $\mathbb {P}_{i,j}$ is by definition the eigenvalue of $A_j$ on the ith weight space $W_i$ which was denoted $\lambda _{m,j,d}$ in previous sections where $m \in V$ is any vector with $Q(m)=i$ . In other words, by definitions, it follows that $\lambda _{m,j,d}=\mathbb {P}_{Q(m),j}$ for all $m \in V, j \in \Delta $ .

In particular, $\mathbb {P}_{\bar {0},t} = \lambda _{0,t,d}=q^d \hat {S}_t(0)=|S_t|$ for all $t \in \Delta $ and thus the top row of $\mathbb {P}$ lists the sphere sizes. Also $\mathbb {P}_{i,\bar {0}} = \lambda _{m,\bar {0},d}=1$ (here, m is any vector with $Q(m)=i$ ). Thus, the leftmost column of $\mathbb {P}$ is all 1’s (eigenvalues of $\mathbb {A}_{\bar {0}}$ which is the identity operator).

On the other hand, if $0 \in \Delta $ , $\mathbb {P}_{0,t} = \lambda _{m,t,d}$ for some nonzero vector m of length $Q(m)=0$ . By Theorem 3.1, $\mathbb {P}_{0,t}=-\delta _{t,0} + q^{\frac {d}{2}-1}\epsilon (q)^d \tilde {K}_d(-t,0)=\mathbb {P}_{\bar {0},t}-q^{d-1}=|S_t|-q^{d-1}$ . Thus, the $0$ -row (which would be listed second if it occurs) consists of the deviations of sphere sizes from the expected size $q^{d-1}$ .

Proposition 4.2 can be recast in terms of the entries of $\mathbb {P}$ as follows: When $i \neq j \in \Delta $ , we have $\sum _{m \in V} \lambda _{m,i,d} \lambda _{m,j,d} = 0$ . Lumping terms in this sum by the length t of the vector m yields

$$ \begin{align*}\sum_{t \in \Delta} |S_t| \mathbb{P}_{t,i} \mathbb{P}_{t,j} = 0. \end{align*} $$

Similarly, we get when $i = j \in \Delta $ , $\sum _{t \in \Delta } |S_t| \mathbb {P}_{t,i}^2 = q^d |S_i|$ .

It follows that $\mathbb {P}^T D \mathbb {P} = q^d D$ , where D is a diagonal matrix with the sizes of the spheres along the diagonal in the same order as they appear in the top row of $\mathbb {P}$ .

As no sphere sizes are zero (we only index distances that occur), D is invertible and it follows that $\mathbb {P}^{-1} = \frac {1}{q^d} D^{-1} \mathbb {P}^T D$ .

As the Q-matrix is defined uniquely (as inverses are unique) by the condition $PQ=q^d I = QP$ , it follows that $Q=D^{-1} \mathbb {P}^T D$ . We summarize this next in the following.

Theorem 5.1 Let q be any odd prime power, $d \geq 2$ and consider the Euclidean association scheme with distance set $\Delta $ . Then $|\Delta |=q+1$ , when $0 \neq \bar {0} \in \Delta $ and $|\Delta |=q$ otherwise when $0 \notin \Delta $ , i.e., when $(d=2$ and $q = 3 \text { mod } 4)$ .

The $|\Delta | \times |\Delta |$ matrices $\mathbb {P}$ and Q of the association scheme (relative to some linear ordering of $\Delta $ for which $\bar {0}$ comes first) have: $\mathbb {P}_{\bar {0},j} = |S_j|$ for all $j \in \Delta $ , $\mathbb {P}_{i,\bar {0}} = 1$ for all $i \in \Delta $ and otherwise for $i, j \in \Delta -\{ \bar {0} \}$ we have

$$ \begin{align*}P_{i,j} = -\delta_{j,0} + q^{\frac{d}{2}-1} \epsilon(q)^d \tilde{K}_d(-j,-\frac{i}{4}), \end{align*} $$

where $\tilde {K}_d(a,b)=\sum _{x \in \mathbb {F}_q^{\times }} \binom {x}{q}^d \chi (ax+\frac {b}{x})$ is a twisted Kloosterman sum and $\epsilon (q)$ is either $1$ or i depending if q is $1$ or $3$ mod $4$ .

If D is the diagonal matrix whose diagonals are the sphere sizes (following the chosen linear ordering of $\Delta $ ), then $Q=D^{-1}\mathbb {P}^T D$ and $Q\mathbb {P}=\mathbb {P}Q=q^d\mathbb {I}$ . Thus $Q_{i,j}=\frac {|S_j|}{|S_i|} P_{j,i}$ .

Note that when $d=2, q=3 \text { mod } 4$ , then $0 \notin \Delta $ and $P_{i,j}=-K(-j,\frac {-i}{4})=-K(1,\frac {ij}{4})$ for any $i, j \in \Delta -\{ \bar {0} \}$ and so other than the first row and column of $\mathbb {P}$ , the other entries are negatives of Kloosterman numbers. In this case, we also have $|S_i|=q+1$ for all $i \neq \bar {0}$ and $|S_{\bar {0}}|=1$ . If we denote $\mathfrak {1}$ to be the ( $q-1$ )-dimensional column vector of all $1$ s, and K to the symmetric $(q-1) \times (q-1)$ matrix with $K_{i,j}=K(1,\frac {ij}{4})=K_{j,i}, i,j \in \mathbb {F}_q^{\times }$ , then we have

$$ \begin{align*}\mathbb{P}=\begin{bmatrix} 1 & (q+1)\mathfrak{1}^T \\ \mathfrak{1} & -K \end{bmatrix}=Q.\end{align*} $$

Thus, $\mathbb {P}^2=q^2\mathbb {I}$ for the Euclidean association scheme when $d=2, q = 3 \text { mod } 4.$

On the other hand, when $d=2, q=1 \text { mod } 4$ , then $0 \in \Delta $ , $|S_i|=q-1$ if $i \neq 0, \bar {0}$ and $|S_0|=2q-2, |S_{\bar {0}}|=1$ . With respect to any linear ordering with $\bar {0} < 0 < \mathbb {F}_q^{\times }$ , the $(q+1) \times (q+1)\ \mathbb {P}$ and Q matrices are given in this case by

$$ \begin{align*}\mathbb{P}=\begin{bmatrix} 1 & 2q-2 & (q-1)\mathfrak{1}^T \\ 1 & q-2 & -\mathfrak{1}^T \\ \mathfrak{1} & (-2)\mathfrak{1} & K \end{bmatrix}\end{align*} $$

and again one can verify that $\mathbb {P}=Q$ using the formulas of the last theorem. Thus again one has $\mathbb {P}^2=q^2\mathbb {I}$ though the dimensions of $\mathbb {P}$ are one larger in this case compared to the last one.

We record these dimension 2 results in the next proposition.

Proposition 5.2 Let $\mathbb {F}_q$ be a finite field of odd characteristic and consider the Euclidean association scheme when dimension $d=2$ . Let $\mathfrak {1}$ to be the ( $q-1$ )-dimensional column vector of all $1$ s, and K be the symmetric $(q-1) \times (q-1)$ matrix with $K_{i,j}=K(1,\frac {ij}{4})=K_{j,i}, i,j \in \mathbb {F}_q^{\times }$ (with respect to some linear ordering of $\mathbb {F}_q^{\times }$ ). Order the distance set $\Delta $ via a linear ordering with $\bar {0} < 0 < \mathbb {F}_q^{\times }$ .

Then if $q=3 \text { mod } 4$ , we have

$$ \begin{align*}\mathbb{P}=\begin{bmatrix} 1 & (q+1)\mathfrak{1}^T \\ \mathfrak{1} & -K \end{bmatrix}=Q\end{align*} $$

are $q \times q$ matrices.

On the other hand, when $q=1 \text { mod } 4$ , we have

$$ \begin{align*}\mathbb{P}=\begin{bmatrix} 1 & 2q-2 & (q-1)\mathfrak{1}^T \\ 1 & q-2 & -\mathfrak{1}^T \\ \mathfrak{1} & (-2)\mathfrak{1} & K \end{bmatrix}=Q\end{align*} $$

are $(q+1) \times (q+1)$ matrices.

In either case, $\mathbb {P}^2=q^2\mathbb {I}$ .

The case when $d \geq 3$ is a bit trickier as the dimension of spheres can vary more. In general, for any $\lambda \in \mathbb {F}_q^{\times }$ , scaling by $\lambda $ gives a bijection between the sphere of radius j about the origin and the sphere of radius $\lambda ^2 j$ about the origin. Thus, there are in general four distinct sizes of spheres $|S_{\hat {0}}|=1$ , $|S_0|$ , $|S_{Sq}|$ and $|S_{Sq^c}|$ , where $|S_{Sq}|$ is the common size of $S_j$ when j is a nonzero square in $\mathbb {F}_q$ and $|S_{Sq^c}|$ is the common size of $S_j$ when j is a nonsquare in $\mathbb {F}_q$ . As these spheres partition $\mathbb {F}_q^d$ , we have

$$ \begin{align*}q^d=1+|S_0| + \frac{q-1}{2}|S_{Sq}| + \frac{q-1}{2} |S_{Sq^c}|\end{align*} $$

for all integers $d \geq 3$ and odd prime powers q.

Pick a primitive generator $\theta $ of the multiplicative group of $\mathbb {F}_q$ (which is a cyclic group of order $q-1$ ). Then $\theta ^j$ is a square in $\mathbb {F}_q$ if and only if j is even and so the Legendre symbol $\binom {\theta ^j}{q}$ is equal to $(-1)^j$ for all integers j. We order the distance set $\Delta =\{ \bar {0} \} \cup \mathbb {F}_q$ via $\bar {0} < 0 < \theta ^0 < \theta ^1 < \theta ^2 < \dots < \theta ^{q-2}$ . Note $4=\theta ^s$ for unique integer s with $0 \leq s \leq q-2$ .

We define the $(q-1) \times (q-1)$ matrix $\tilde {\mathbb {K}}_d$ (whose rows and columns are indexed by $\{0,1,\dots , q-2\}$ in the natural order) via

$$ \begin{align*} (\tilde{\mathbb{K}}_d)_{i,j} &= \epsilon(q)^d \tilde{K}_d(-\theta^j, -\frac{\theta^i}{4}) \\ &=\epsilon(q)^d \binom{-\theta^j}{q}^d \tilde{K}_d(1,\frac{\theta^i\theta^j}{4})\\ &=\epsilon(q)^d\binom{-1}{q}^d (-1)^{jd}\tilde{K}_d(1,\theta^{i+j-s}), \end{align*} $$

where $\tilde {K}(a,b)$ are the (twisted) Kloosterman sums. The second equality follows from part (2) of Proposition 4.1.

This matrix is always a real matrix from the earlier discussion about twisted Kloosterman sums. When d is even, it is symmetric and circulant. Recall, a circulant matrix is one such that each row of the matrix is the cyclic shift of the row above it, one notch to the left. When d is odd, it is no longer circulant or symmetric but is signed-circulant, i.e., each row is the negative of the cyclic shift of the row above it, one notch to the left.

Using these conventions and Theorem 5.1, the following proposition follows by a simple computation with the help of the identities in Proposition 4.1.

Proposition 5.3 Let $\mathbb {F}_q$ be a finite field of odd characteristic and consider the Euclidean association scheme of dimension $d \geq 3$ . Fix a primitive generator $\theta $ of $\mathbb {F}_q^{\times }$ , then $4=\theta ^s$ for unique integer $0 \leq s \leq q-2$ .

Let $(\tilde {\mathbb {K}}_d)_{i,j} =\epsilon (q)^d\binom {-1}{q}^d (-1)^{jd}\tilde {K}_d(1,\theta ^{i+j-s})$ whose rows and columns are indexed by $\{0,\dots , q-2\}$ . It is a real, symmetric, and circulant matrix when d is even and it is a real, signed-circulant matrix when d is odd.

Let $\mathfrak {1}$ be the $(q-1)$ -dimensional column vector of all ones, and let $\hat {\eta }$ be the $(q-1)$ -dimensional row vector whose entries alternate between $|S_{Sq}|$ and $|S_{Sq^c}|$ , starting with the former. Let $\hat {\mu }$ be the $(q-1)$ -dimensional column vector whose entries are $(-1)^{id}$ as i ranges from $0$ to $q-2$ . (So it always starts at $1$ and alternates sign if d is odd, but is the all one vector when d is even.)

With respect to the ordering $\bar {0} < 0 < \theta ^0 < \theta ^1 < \theta ^2 < \dots < \theta ^{q-2}$ of the distance set $\Delta $ , the P matrix of the Euclidean association scheme is a $(q+1)\times (q+1)$ matrix given by

$$ \begin{align*}\mathbb{P}=\begin{bmatrix} 1 & |S_0| & \hat{\eta} \\ 1 & \alpha & \beta\hat{\mu}^T \\ \mathfrak{1} & \beta\hat{\mu}-\mathfrak{1} & q^{\frac{d}{2}-1}\tilde{\mathbb{K}}_d \end{bmatrix}\end{align*} $$

where $\alpha =-1+q^{\frac {d}{2}-1}\epsilon (q)^d (q-1)\frac {(-1)^d+1}{2}$ and $\beta =-q^{\frac {d}{2}-1}\epsilon (q)^d$ when d even, $\beta =\binom {-1}{q} q^{\frac {d-1}{2}}\epsilon (q)^{d+1}$ when d odd.

The Q matrix has $Q_{i,j}=\frac {|S_j|}{|S_i|}P_{j,i}$ . Due to varying sizes of spheres when $d \geq 3$ , one no longer has $Q=P$ in general.

The above proposition also works for the $d=2$ case though in the case $q=3 \text { mod } 4$ one should throw out the second row and column as the distance $0$ does not occur.

6 Equidistribution

Through work of Kloosterman and A. Weil, for any odd prime power q and $a \in \mathbb {F}_q^{\times }$ , it is known that the Kloosterman numbers $K_q(1,a)=\sum _{x \in \mathbb {F}_q^{\times }} \chi (x + \frac {a}{x})$ satisfy

$$ \begin{align*}K_q(1,a)=\sqrt{q}(e^{i \theta_{q,a}} + e^{-i \theta_{q,a}})=2\sqrt{q}\cos(\theta_{q,a}) \end{align*} $$

for a unique “Kloosterman” angle $\theta _{q,a} \in [0, \pi ]$ .

Using the sophisticated tools of lisse sheafs and etale cohomology, N. Katz (slightly simplified proofs later by Adolphson), proved the “vertical equidistribution” of these numbers that states that as $q \to \infty $ , the distribution of these angles approaches the Sato–Tate measure on $[0,\pi ]$ . We discuss this more carefully next.

Definition 6.1 (Sato–Tate measure)

The Sato–Tate Borel probability measure $\mu _{ST}$ on $[0,\pi ]$ is given by the condition

$$ \begin{align*}\mu_{ST}([a,b]) = \frac{2}{\pi} \int_a^b sin^2(\theta) d\theta \end{align*} $$

for all $0\leq a < b \leq \pi $ .

$\mu _{ST}$ is characterized as the unique Borel measure being absolutely continuous with respect to Lebesgue measure with Radon–Nikodym derivative $\frac {2}{\pi } sin^2(\theta )$ .

For any continuous $f: [0, \pi ] \to \mathbb {R}$ , we write

$$ \begin{align*}E_{ST}[f] = \frac{2}{\pi} \int_0^{\pi} f(\theta) sin^2(\theta) d\theta\end{align*} $$

for the expectation of f with respect to this probability measure.

Definition 6.2 (Kloosterman angle average)

For any odd prime power q, let $\theta _{a,q}$ be the Kloosterman angles associated with the Kloosterman sums $K(1,a), a \in \mathbb {F}_q^{\times }$ . Given a continuous function $f: [0, \pi ] \to \mathbb {R}$ , we let

$$ \begin{align*}E_{K,q}[f] = \frac{1}{q-1} \sum_{a \in \mathbb{F}_q^{\times}} f(\theta_{q,a}) \end{align*} $$

be the “sample average” of f over these Kloosterman angles.

We are now ready to state the deep vertical equidistribution theorem of N. Katz (see [Reference Katz10]).

Theorem 6.3 (Vertical equidistribution of Kloosterman sums)

For any sequence of odd prime powers $q_n \to \infty $ , and any continuous function $f: [0, \pi ] \to \mathbb {R}$ , we have

$$ \begin{align*}\lim_{n \to \infty} E_{K,q_n}[f] = E_{ST}[f]. \end{align*} $$

Next, we will discuss some graph theoretical equivalents to the equidistribution theorem. First, some basic trigonometric facts will be collected.

Proposition 6.4 For any nonnegative integers $m, n,$ we have:

(1) $\frac {2}{\pi } \int _{0}^{\pi } \cos (m \theta )\cos (n \theta )d\theta = \delta _{m,n}+\delta _{m,n}\delta _{m,0}.$

(2) $E_{ST}[\cos (n\theta )]=0$ if $n \neq 0,2$ . $E_{ST}[\cos (2\theta )]=-\frac {1}{2}$ .

The Sato–Tate probability measure is characterized by these expectations amongst continuous probability measures on $[0, \pi ]$ .

(3) For every $\ell \geq 1$ , we have

$$ \begin{align*}2^{2\ell-1}\cos^{2\ell}(\theta)=\sum_{k=0}^{\ell-1} \binom{2\ell}{k} \cos((2\ell-2k)\theta) + \frac{1}{2} \binom{2\ell}{\ell}.\end{align*} $$

(4) For every $\ell \geq 1$ , we have

$$ \begin{align*}2^{2\ell}\cos^{2\ell+1}(\theta)=\sum_{k=0}^{\ell} \binom{2\ell+1}{k} \cos((2\ell+1-2k)\theta).\end{align*} $$

(5) For positive integer m, we have $E_{ST}[\cos ^{2m+1}(\theta )]=0$ and

$$ \begin{align*}E_{ST}[cos^{2m}(\theta)]=\frac{1}{2^{2m-1}} (\frac{-1}{2} \binom{2m}{m-1} + \frac{1}{2} \binom{2m}{m})=\frac{1}{2^{2m}(m+1)}\binom{2m}{m}.\end{align*} $$

Proof Proof of (1): Follows from the trigonometric identity

$$ \begin{align*}\cos(mx)\cos(nx) = \frac{1}{2}\cos((m+n)x) + \frac{1}{2}\cos((m-n)x) \end{align*} $$

and simple integration.

Proof of (2): Follows from (1), once one notes that $\sin ^2(\theta )=\frac {1-\cos (2\theta )}{2}$ . The characterization of Sato–Tate measure from these expectations follows from the density of the algebra generated by the $cos(n\theta )$ in the ring of continuous real-valued functions on $[0,\pi ]$ (with the supremum metric) which itself follows from a generalized Stone–Weierstrass theorem, together with the Riesz representation theorem.

Proof of (3): Write $2\cos (\theta )=e^{i\theta } + e^{-i\theta }$ and raise both sides to the $2\ell $ power to conclude

$$ \begin{align*}2^{2\ell} \cos^{2\ell}(\theta)=\sum_{k=0}^{2\ell} \binom{2\ell}{k} (e^{i\theta})^{2\ell-k} (e^{-i\theta})^k = \sum_{k=0}^{2\ell} \binom{2\ell}{k} (e^{i\theta})^{2\ell-2k}. \end{align*} $$

Finish by grouping the $k=j$ and $k=2\ell -j$ terms for each j.

Proof of (4): Follows the same procedure as for (3).

Proof of (5): Follows from using (3), (4) in (2).

Definition 6.5 (Kloosterman moments)

For any odd prime power q, and positive integer $\ell $ , let

$$ \begin{align*}M_{q,\ell} = \sum_{a \in \mathbb{F}_q^{\times}} K_q(1,a)^\ell = 2^{\ell} q^{\frac{\ell}{2}}(q-1) E_{K,q}[cos^{\ell}(\theta)] \end{align*} $$

be the $\ell $ th Kloosterman moment.

Using little-oh notation and the trigonometric identities in Proposition 6.4, we find that the vertical equidistribution of Kloosterman sums is equivalent to establishing the following behavior for Kloosterman moments:

$$ \begin{align*}M_{q, 2\ell+1} = o(q^{\ell+1.5}) \end{align*} $$

and

$$ \begin{align*}M_{q,2\ell} = q^{\ell+1} \frac{1}{\ell+1}\binom{2\ell}{\ell} (1 + o(1)) \end{align*} $$

as $q \to \infty $ .

The quantity $C_{\ell }=\frac {1}{\ell +1}\binom {2\ell }{\ell }$ in the last limit is the $\ell $ th Catalan number which occurs frequently in combinatorics. It is, for example, the number of ways to bracket a given $(\ell +1)$ -fold product in terms of pairwise multiplications.

Recall, in a graph, two edges are incident if they share a common vertex and a walk of length $\ell $ is a sequence of $\ell $ edges, where each edge is incident to the previous one. If $\mathbb {A}$ is the adjacency matrix of the graph with respect to some vertex ordering, then the $(v,w)$ -entry of $\mathbb {A}^{\ell }$ is the number of walks of length $\ell $ from vertex v to vertex w in the graph. Such a walk is called closed if $v=w$ . Thus, $\mathrm{Trace}(\mathbb {A}^{\ell })$ is the number of closed walks of length $\ell $ in the graph.

Let q be an odd prime power, $q= 3 \text { mod } 4$ . Fix any $t \in \mathbb {F}_q^{\times }$ then we have seen that the spectrum of $\mathbb {A}_t$ , the adjacency matrix of the distance-t graph for the plane $\mathbb {F}_q^2$ is the multiset

$$ \begin{align*}\{\{ (q+1)^{(1)}, -K_q(1,a)^{(q+1)} | a \in \mathbb{F}_q^{\times} \} \},\end{align*} $$

where the superscripts indicate the multiplicity of each eigenvalue in the multiset.

Thus

$$ \begin{align*}\mathrm{Trace}(\mathbb{A}_t^{\ell}) = (q+1)^{\ell} + (q+1)(-1)^{\ell}\sum_{a \in \mathbb{F}_q^{\times}} K_q(1,a)^{\ell} = (q+1)^{\ell} + (q+1)(-1)^{\ell}M_{q,\ell}. \end{align*} $$

On the other hand, when $q=1 \text { mod } 4$ , the spectrum of $\mathbb {A}_t$ is

$$ \begin{align*}\{\{ (q-1)^{(1)}, (-1)^{(2q-2)}, K_q(1,a)^{(q-1)} | a \in \mathbb{F}_q^{\times} \} \}\end{align*} $$

and so

$$ \begin{align*}\mathrm{Trace}(\mathbb{A}_t^{\ell}) &= (q-1)^{\ell} + (2q-2)(-1)^{\ell} + (q-1)\sum_{a \in \mathbb{F}_q^{\times}} K_q(1,a)^{\ell}\\ &= (q-1)^{\ell} + (2q-2)(-1)^{\ell} + (q-1)M_{q,\ell}. \end{align*} $$

We record these results in the next proposition.

Proposition 6.6 Let q be an odd prime power, and let $t \in \mathbb {F}_q^{\times }$ and $\mathbb {A}_t$ be the adjacency matrix of the distance-t graph for the plane $\mathbb {F}_q^2$ . Then if $q = 3 \text { mod } 4$ , we have the number of closed walks of length $\ell $ is

$$ \begin{align*}\mathrm{Trace}(\mathbb{A}_t^{\ell}) = (q+1)^{\ell} + (q+1)(-1)^{\ell}M_{q,\ell}. \end{align*} $$

If $q=1 \text { mod } 4$ , we have

$$ \begin{align*}\mathrm{Trace}(\mathbb{A}_t^{\ell}) = (q-1)^{\ell} + (2q-2)(-1)^{\ell} + (q-1)M_{q,\ell,} \end{align*} $$

where $M_{q,\ell }$ is the $\ell $ th Kloosterman moment.

Vertical equidistribution of Kloosterman angles is equivalent to

$$ \begin{align*}M_{q, 2\ell+1} = o(q^{\ell+1.5}) \end{align*} $$

and

$$ \begin{align*}M_{q,2\ell} = q^{\ell+1} \frac{1}{\ell+1}\binom{2\ell}{\ell} (1 + o(1)) \end{align*} $$

as $q \to \infty $ .

7 An application to the random walk in finite planes of odd order

Fix $t \neq 0$ in $\mathbb {F}_q$ and let $\mathbb {A}_t$ be the adjacency matrix for the distance-t graph in the plane $\mathbb {F}_q^2$ . Define $T_t=\frac {1}{|S_t|} \mathbb {A}_t$ to be the transition matrix of the corresponding random walk Markov chain. The $(i,j)$ -entry of $T_t^{\ell }$ represents the probability of transitioning from the ith vertex to the jth vertex after $\ell $ steps in this Markov chain. As the $A_t$ -graph is vertex transitive, the diagonal entries of $T_t^{\ell }$ are all equal as the probability that we return to vertex i after $\ell $ steps, if we start at vertex i, will be independent of the vertex. Thus $(T_t^{\ell })_{i,i}=\frac {1}{q^2}\mathrm{Trace}(T_t^{\ell })$ for any i.

Using Bayesian conditioning, it follows that for any initial probability distribution $p_i$ on the vertices, the chance that you start and end at the same vertex after $\ell $ steps is $ \sum _{j=1}^{q^2} (T_t^{\ell })_{j,j}p_j= \frac {1}{q^2} \mathrm{Trace} (T_t^{\ell })\sum _{j=1}^{q^2} p_j =\frac {1}{q^2|S_t|^{\ell }} \mathrm{Trace}(A_t^{\ell })$ is the probability that you return to where you start after $\ell $ steps (it is independent of initiate state). Let us call this quantity the probability of return and denote it $R_{q,\ell ,t}$ . Now $|S_t|=q \pm 1$ where the sign depends if q is $1$ or $3$ mod $4$ so we find that this probability is $R_{q,\ell ,t}=\frac {1}{q^2(q \pm 1)^{\ell }} \mathrm{Trace}(A_t^{\ell })$ .

By Corollary 6, we see that if $q=3 \text { mod } 4$ , and positive $\ell $ , we have

$$ \begin{align*}R_{q,\ell,t}=R_{q,\ell}=\frac{1}{q^2}(1 + \frac{(-1)^{\ell}}{(q+1)^{\ell-1}} M_{q,\ell}) \end{align*} $$

does not depend on $t \neq 0$ . Note the first $\frac {1}{q^2}$ term is what one would expect if the location after $\ell $ steps were equally likely to be anywhere in the plane and the second term represents an arithmetic bias against that happening. Vertical equidistribution of Kloosterman sums, tells us furthermore that

$$ \begin{align*}R_{q, 2\ell} = \frac{1}{q^2} + \frac{q^{\ell-1}}{(\ell+1)(q+1)^{2\ell-1}}\binom{2\ell}{\ell} (1 + o(1)) = \frac{1}{q^2} + \frac{1}{q^{\ell}(\ell+1)}\binom{2\ell}{\ell}(1+o(1)) \end{align*} $$

and

$$ \begin{align*}R_{q, 2\ell+1}=\frac{1}{q^2}(1 - \frac{1}{(q+1)^{2\ell}}o(q^{\ell+1.5}))=\frac{1}{q^2}(1+o(q^{1.5-\ell})) \end{align*} $$

as $q \to \infty $ through such prime powers.

Thus we have proven:

Theorem 7.1 (Probability of return in the distance-t random walk in $\mathbb {F}_q$ -planes)

Let q be an odd prime, $q=3 \text { mod } 4$ . Let $R_{q,\ell ,t}$ be the probability that you return to the same vertex after $\ell $ steps in the distance-t walk where $t \neq 0$ . Then $R_{q,\ell , t}=R_{q,\ell }$ is independent of $t \neq 0$ and initial state. We have

$$ \begin{align*}R_{q,\ell,t}=R_{q,\ell}=\frac{1}{q^2}(1 + \frac{(-1)^{\ell}}{(q+1)^{\ell-1}} M_{q,\ell}). \end{align*} $$

Furthermore, as $q \to \infty $ , we have

$$ \begin{align*}R_{q, 2\ell} = \frac{1}{q^2} + \frac{q^{\ell-1}}{(\ell+1)(q+1)^{2\ell-1}}\binom{2\ell}{\ell} (1 + o(1)) = \frac{1}{q^2} + \frac{1}{q^{\ell}(\ell+1)}\binom{2\ell}{\ell}(1+o(1)) \end{align*} $$

and

$$ \begin{align*}R_{q, 2\ell+1}=\frac{1}{q^2}(1 - \frac{1}{(q+1)^{2\ell}}o(q^{\ell+1.5}))=\frac{1}{q^2}(1+o(q^{1.5-\ell})). \end{align*} $$

8 Intersection numbers and matrices of the scheme

In this section, we calculate the intersection numbers $p_{i,j}^k$ of the d-dimensional Euclidean association scheme. Recall $p_{i,j}^k=p_{j,i}^k$ is the number of ways a pair of points $\{x,y\}$ in $\mathbb {F}_q^d$ with $d(x,y)=k$ can be completed to a triangle $\{x,y,z\}$ of side lengths $i,j,k$ . Note $p_{i,j}^{\bar {0}} =\delta _{i,j} |S_i|$ and $p_{\bar {0},j}^k =\delta _{j,k}$ so we may assume $i,j,k \in \mathbb {F}_q$ and so are not $\bar {0}$ .

We will concentrate on the planar, $d=2$ case as the higher-dimensional cases were reduced to this case in [Reference Kwok12]. This case was partially computed in [Reference Bennet, Iosevich and Pakianathan3] in a different context.

Theorem 8.1 (Intersection numbers of planar Euclidean association scheme)

Let q be an odd prime power, and let $p_{i,j}^k$ be the intersection numbers of the planar ( $d=2$ ) Euclidean association scheme for $i,j,k \in \mathbb {F}_q$ not equal to $\bar {0}$ .

If $q=1 \text { mod } 4$ , we have $p_{i,j}^0=1+(q-2)\delta _{i,j}\delta _{i,0}-\delta _{i,j}$ .

Otherwise, $k \neq 0$ and we have

$$ \begin{align*}p_{i,j}^k=\binom{4\sigma_2-\sigma_1^2}{q}+1, \end{align*} $$

where $\binom {x}{q}$ is the usual Legendre symbol, $\sigma _1=i+j+k, \sigma _2=ij+jk+ki$ .

Furthermore, when $i,j,k \in \mathbb {F}_q$ are the distances in a triple of points of $\mathbb {F}_q^2$ , we have $4\sigma _2 = \sigma _1^2$ if and only if $\{i,j,k\}$ are the distances of a collinear triple of points (a triple of points that lies on an affine line).

We record the following corollary, which exploits the odd behavior of isotropic lines when they exist.

We now recall the definition of the intersection matrices of an association scheme.

Definition 8.3 (Intersection matrices)

Let $(V,d)$ be an association scheme with distance set $\Delta $ and intersection numbers $p_{i,j}^k$ . We define $|\Delta |$ many $|\Delta | \times |\Delta |$ intersection matrices of the scheme $L_i, i \in \Delta $ via

$$ \begin{align*}(L_i)_{k,j} = p_{i,j}^k.\end{align*} $$

In general, these satisfy $L_{\bar {0}}=\mathbb {I}$ and $L_i L_j = \sum _{k \in \Delta } p_{i,j}^k L_k$ . Thus the map

$$ \begin{align*}\mathbb{R}[A_i, i \in \Delta] \to \mathbb{R}[L_i, i \in \Delta]\end{align*} $$

is an algebra epimorphism from the Bose–Messner algebra to the real algebra generated by these intersection matrices. Note the dimensions $|V| \times |V|$ of the $A_i$ matrices are in general quite different than the size of the $L_i$ matrices.

The eigenvalues of the $L_i$ matrix are the same as that of the $A_i$ matrix but with different multiplicities. In fact the columns of the scheme’s Q matrix are the corresponding simultaneous (right) eigenvectors of the $L_i$ while the rows of the P matrix are the (left) eigenvectors of the $L_i$ . These facts follow from the identity

$$ \begin{align*}PL_jP^{-1}=\mathrm{diag}(P_{\bar{0},j},\dots, P_{d,j})\end{align*} $$

(see section 11.2 of [Reference Brouwer and Haemers5]).

We now record the intersection matrices for the planar Euclidean association scheme.

Theorem 8.4 Let q be an odd prime power and consider the planar Euclidean association scheme on $\mathbb {F}_q^2$ . Note $L_{\bar {0}}$ is always an identity matrix. Recall when $q=1 \text { mod } 4$ , we have $\bar {0} \neq 0 \in \Delta $ also.

When $i,j \in \mathbb {F}_q^{\times }$ , we have $(L_i)_{\bar {0},j}=p_{i,j}^{\bar {0}}=\delta _{i,j}|S_i|=\delta _{i,j}(q-\epsilon _q^2)$ and $(L_i)_{0,j}=p_{i,j}^{0}=1-\delta _{i,j}$ .

We have $(L_i)_{j,\bar {0}}=p_{i,\bar {0}}^j=\delta _{i,j}$ and $(L_i)_{\bar {0},\bar {0}}=p^{\bar {0}}_{i,\bar {0}}=0$ .

When $i,j \in \mathbb {F}_q$ , $k \in \mathbb {F}_q^{\times }$ , we have

$$ \begin{align*}(L_i)_{k,j}=p^k_{i,j}=\binom{4\sigma_2 - \sigma_1^2}{q} + 1 \in \{0,1,2\}, \end{align*} $$

where $\sigma _2=ij+jk+ki, \sigma _1=i+j+k$ and $\binom {x}{q}$ is the Legendre symbol.

When $q=3 \text { mod } 4$ , the spectrum of $L_i$ is $\{\{ (q+1)^{(1)}, -K_q(1,a)^{(1)}, a \in \mathbb {F}_q^{\times } \}\}$ for any $i \in \mathbb {F}_q^{\times }$ . Thus $\mathrm{Trace}(L_i^{\ell })=(q+1)^{\ell } + (-1)^{\ell }M_{q,\ell }$ .

When $q=1 \text { mod } 4$ , the spectrum of $L_i$ is $\{\{ (q-1)^{(1)}, -1^{(1)}, K_q(1,a)^{(1)}, a \in \mathbb {F}_q^{\times } \}\}$ for any $i \in \mathbb {F}_q^{\times }$ . Thus $\mathrm{Trace}(L_i^{\ell })=(q-1)^{\ell } + (-1)^{\ell } + M_{q,\ell }$ .

Note that the entries of $L_i$ keep track of which triangles exist in the plane $\mathbb {F}_q^2$ and through the last theorem, implicitly determine the Kloosterman moments $M_{q,\ell }$ for all $\ell $ . As it is well known that a finite multiset of complex numbers is determined by all its moments, this means the Kloosterman sums as a set are determined by exactly the data of which triangles exist in the plane $\mathbb {F}_q^2$ .