A structure theorem for product sets in extra special groups

TL;DR

This paper provides a new, robust proof of a structure theorem for product sets in extra special groups, extending previous results from Heisenberg groups to a broader class of groups and quasigroups.

Contribution

The authors introduce a new proof technique that generalizes the structure theorem for product sets from Heisenberg groups to all extra special groups and certain quasigroups.

Findings

Product sets in large bricks of extra special groups contain many cosets of the center.

The new proof extends the theorem's applicability beyond Heisenberg groups.

The approach is robust and adaptable to a wider class of algebraic structures.

Abstract

Hegyv\'ari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $B\cdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.

Equations

$$H_n=⟨$$

$$[a_i,c]=[b_i,c]=1,[a_i,b_i]=c,a_i^p=b_i^p=c_i^p=1\text{mbox}for1\le i\le n⟩$$

$$M_n=⟨$$

$$[a_i,c]=[b_i,c]=1,[a_i,b_i]=c,a_i^p=c_i^p=1,b_i^p=c\text{mbox}for1\le i\le n⟩.$$

$$[\underline{x},\underline{y},z]:=\left[\begin{array}{ccc}1& \underline{x}& z\\ 0& I_n& \underline{y}\\ 0& 0& 1\end{array}\right]$$

$$[\underline{x},\underline{y},z]\cdot [{\underline{x}}^{\prime },{\underline{y}}^{\prime },z^{\prime }]=[\underline{x}+{\underline{x}}^{\prime },\underline{y}+{\underline{y}}^{\prime },z+z^{\prime }+⟨\underline{x},{\underline{y}}^{\prime }⟩]$$

$$B=\{[\underline{x},\underline{y},z]\text{ such that }\underline{x}\in \underline{X},\underline{y}\in \underline{Y},z\in Z\}$$

$$|B|>|H_n{|}^{3/4+\epsilon }$$

$$\{[\underline{x},\underline{y},z]:\underline{x}\in Q^n,\underline{y}\in Q^n,z\in Q\}$$

$$[\underline{x},\underline{y},z]\cdot [{\underline{x}}^{\prime },{\underline{y}}^{\prime },z^{\prime }]=[\underline{x}+{\underline{x}}^{\prime },\underline{y}+{\underline{y}}^{\prime },z+z^{\prime }+⟨\underline{x},{\underline{y}}^{\prime }⟩],$$

$$|B|>|M_n{|}^{3/4+\epsilon }$$

$$|B|>|G{|}^{3/4+\epsilon }$$

$$B=\{[\underline{x},\underline{y},z]\text{ such that }\underline{x}\in \underline{X},\underline{y}\in \underline{Y},z\in Z\}$$

$$|B|>|H_n(Q){|}^{3/4+\epsilon },$$

$$[\underline{x},\underline{y},z]\cdot [{\underline{x}}^{\prime },{\underline{y}}^{\prime },z^{\prime }]=[\underline{x}+{\underline{x}}^{\prime },\underline{y}+{\underline{y}}^{\prime },z+z^{\prime }+⟨\underline{x},{\underline{y}}^{\prime }⟩+f(\underline{y},{\underline{y}}^{\prime })]$$

$$f\left((y_1,\dots ,y_n),(y_1^{\prime },\dots ,y_n^{\prime })\right)=\sum _{i=1}^n\lfloor \frac{y_i\mathrm{mod}p+y_i^{\prime }\mathrm{mod}p}{p}\rfloor ,$$

$$f(\underline{y}+{\underline{y}}^{\prime },{\underline{y}}^{\prime \prime })+f(\underline{y},{\underline{y}}^{\prime })=f(\underline{y},{\underline{y}}^{\prime }+{\underline{y}}^{\prime \prime })+f({\underline{y}}^{\prime },{\underline{y}}^{\prime \prime })$$

$$\lfloor \frac{(y_i+y_i^{\prime })\mathrm{mod}p+y_i^{\prime \prime }\mathrm{mod}p}{p}\rfloor +\lfloor \frac{y_i\mathrm{mod}p+y_i^{\prime }\mathrm{mod}p}{p}\rfloor$$

$$=\lfloor \frac{y_i\mathrm{mod}p+(y_i^{\prime }+y_i^{\prime \prime })\mathrm{mod}p}{p}\rfloor +\lfloor \frac{y_i^{\prime }\mathrm{mod}p+y_i^{\prime \prime }\mathrm{mod}p}{p}\rfloor .$$

$$\lfloor \frac{y_i\mathrm{mod}p+y_i^{\prime }\mathrm{mod}p+y_i^{\prime \prime }\mathrm{mod}p}{p}\rfloor \in \{0,1,2\}$$

$$y_i\mathrm{mod}p+y_i^{\prime }\mathrm{mod}p+y_i^{\prime \prime }\mathrm{mod}p.$$

$$B=\{[\underline{x},\underline{y},z]\text{ such that }\underline{x}\in \underline{X},\underline{y}\in \underline{Y},z\in Z\}$$

$$\left|e(B,C)-\frac{d|B||C|}{n}\right|\le {\lambda }_2\sqrt{|B||C|}.$$

$$|Z{|}^2\prod _{i=1}^n|X_i\cap (a_i-X_i)||Y_i\cap (b_i-Y_i)|>2p^{n+2},$$

$$B\cdot B\supseteq [\underline{a},\underline{b},{\mathbb{F}}_p].$$

$$B\cdot B\supseteq \{[\underline{x},\underline{y},z]\cdot [\underline{a}-\underline{x},\underline{b}-\underline{y},z^{\prime }]:\underline{x}\in X^{\prime },\underline{y}\in Y^{\prime },z,z^{\prime }\in Z\}.$$

$$[\underline{x},\underline{y},z]\cdot [\underline{a}-\underline{x},\underline{b}-\underline{y},z^{\prime }]=[\underline{a},\underline{b},z+z^{\prime }+⟨\underline{x},(\underline{b}-\underline{y})⟩+f(\underline{y},\underline{b}-\underline{y})].$$

$$\left\{z+z^{\prime }+⟨\underline{x},(\underline{b}-\underline{y})⟩+f(\underline{y},\underline{b}-\underline{y}):z,z^{\prime }\in Z,\underline{x}\in X^{\prime },\underline{y}\in Y^{\prime }\right\}={\mathbb{F}}_p$$

$$E=X^{\prime }\times (-Z+\lambda ),F=\left\{(\underline{b}-\underline{y},-z-f(\underline{y},\underline{b}-\underline{y})):z\in Z,\underline{y}\in Y^{\prime }\right\}.$$

$$z+z^{\prime }+⟨\underline{x},(\underline{b}-\underline{y})⟩+f(\underline{y},\underline{b}-\underline{y})=\lambda .$$

$$\left\{z+z^{\prime }+⟨\underline{x},(\underline{b}-\underline{y})⟩+f(\underline{y},\underline{b}-\underline{y}):z,z^{\prime }\in Z,\underline{x}\in X^{\prime },\underline{y}\in Y^{\prime }\right\}={\mathbb{F}}_p.\text{qed}$$

$$B\cdot B=[2\underline{X},2\underline{Y},{\mathbb{F}}_p].$$

Full text

1 Introduction

Let $p$ be a prime. An extra special group $G$ is a $p$-group whose center $Z$ is cyclic of order $p$ such that $G/Z$ is an elementary abelian $p$-group (nice treatments of extra special groups can be found in [2, 6]). The extra special groups have order $p^{2n+1}$ for some $n\geq 1$ and occur in two families. Denote by $H_{n}$ and $M_{n}$ the two non-isomorphic extra special groups of order $p^{2n+1}$. Presentations for these groups are given in [4]:

[TABLE]

From these presentations, it is not hard to see that the center of each of these groups consists of the powers of $c$ so are cyclic of order $p$. It is also clear that the quotient of both groups by their centers yield elementary abelian $p$-groups.

In this paper we consider the structure of products of subsets of extra special groups. The structure of sum or product sets of groups and other algebraic structures has a rich history in combinatorial number theory. One seminal result is Freiman’s theorem [5], which asserts that if $A$ is a subset of integers and $|A+A|=O(|A|)$, then $A$ must be a subset of a small generalized arithmetic progression. Green and Ruzsa [7] showed that the same result is true in any abelian group. On the other hand, commutativity is important as the theorem is not true for general non-abelian groups [8]. With this in mind, Hegyvári and Hennecart were motivated to study what actually can be said about the structure of product sets in non-abelian groups.

The group $H_{1}$ is the classical Heisenberg group, so the groups $H_{n}$ form natural generalizations of the Heisenberg group. The group $H_{n}$ has a well-known representation as a subgroup of $\operatorname{GL}_{n+2}(\mathbb{F}_{p})$ consisting of upper triangular matrices

[TABLE]

where $\underline{x},\underline{y}\in\mathbb{F}_{p}^{n}$, $z\in\mathbb{F}_{p}$, and $I_{n}$ is the $n\times n$ identity matrix. Let $\underline{e_{i}}\in\mathbb{F}_{p}^{n}$ be the $i\textsuperscript{th}$ standard basis vector. In the presentation for $H_{n}$, $a_{i}$ corresponds to $[\underline{e_{i}},0,0]$, $b_{i}$ corresponds to $[0,\underline{e_{i}},0]$ and $c$ corresponds to $[0,0,1]$. By matrix multiplication, we have

[TABLE]

where $\langle\,,\rangle$ denotes the usual dot product.

Let $H_{n}$ be a Heisenberg group. A subset $B$ of $H_{n}$ is said to be a brick if

[TABLE]

where $\underline{X}=X_{1}\times\dots\times X_{n}$ and $\underline{Y}=Y_{1}\times\dots\times Y_{n}$ with non empty-subsets $X_{i},Y_{i},Z\subseteq\mathbb{F}_{p}$.

Theorem 1.1** (Hegyvári-Hennecart, [9]).**

For every $\varepsilon>0,$ there exists a positive integer $n_{0}$ such that if $n\geq n_{0}$, $B\subseteq H_{n}$ is a brick and

[TABLE]

*then there exists a non trivial subgroup $G^{\prime}$ of $H_{n}$, namely its center $[\underline{0},\underline{0},\mathbb{F}_{p}]$, such that $B\cdot B$ contains at least $|B|/p$ cosets of $G^{\prime}$. *

Our main focus is to extend this theorem to the second family of extra special groups $M_{n}$. A second focus of this paper is to consider generalizations of the higher dimensional Heisenberg groups where entries come from a (left) quasifield $Q$ rather than $\mathbb{F}_{p}$. We recall the definition of a (left) quasifield:

A set $L$ with a binary operation $*$ is called a loop if

the equation $a*x=b$ has a unique solution in $x$ for every $a,b\in L$, 2. 2.

the equation $y*a=b$ has a unique solution in $y$ for every $a,b\in L$, and 3. 3.

there is an element $e\in L$ such that $e*x=x*e=x$ for all $x\in L$.

A (left) quasifield $Q$ is a set with two binary operations $+$ and $*$ such that $(Q,+)$ is a group with additive identity 0, $(Q^{*},*)$ is a loop where $Q^{*}=Q\backslash\{0\}$, and the following three conditions hold:

$a*(b+c)=a*b+a*c$ for all $a,b,c\in Q$, 2. 2.

$0*x=0$ for all $x\in Q$, and 3. 3.

the equation $a*x=b*x+c$ has exactly one solution for every $a,b,c\in Q$ with $a\neq b$.

Throughout the rest of the paper we will use the term quasifield to mean left quasifield. Given a quasifield $Q$, we define $H_{n}(Q)$ by the set of elements

[TABLE]

and a multiplication operation defined by

[TABLE]

where for $\underline{x},\underline{y}\in Q^{n}$, if $\underline{x}=(x_{1},\cdots,x_{n})$ and $\underline{y}=(y_{1},\cdots,y_{n})$, we define $\langle\underline{x},\underline{y}\rangle=\sum_{i=1}^{n}x_{i}*y_{i}$. Then $H_{n}(Q)$ is a quasigroup with an identity element (ie, a loop), and when $Q=\mathbb{F}_{p}$ we have that $H_{n}(Q)$ is the $n$-dimensional Heisenberg group $H_{n}$.

1.1 Statement of main results

Our theorems are analogous to Hegyvári and Hennecart’s theorem for the groups $M_{n}$ and the quasigroups $H_{n}(Q)$. In particular, their structure result is true for all extra special groups. We will define what it means for a subset $B$ of $M_{n}$ to be a brick in Section 2.1.

Theorem 1.2**.**

For every $\varepsilon>0$, there exists a positive integer $n_{0}=n_{0}(\varepsilon)$ such that if $n\geq n_{0}$, $B\subseteq M_{n}$ is a brick and

[TABLE]

then there exists a non trivial subgroup $G^{\prime}$ of $M_{n}$, namely its center, such that $B\cdot B$ contains at least $|B|/p$ cosets of $G^{\prime}$.

Combining Theorem 1.1 and Theorem 1.2, we have

Theorem 1.3**.**

Let $G$ be an extra special group. For every $\varepsilon>0,$ there exists a positive integer $n_{0}=n_{0}(\epsilon)$ such that if $n\geq n_{0}$, $B\subseteq G$ is a brick and

[TABLE]

then there exists a non trivial subgroup $G^{\prime}$ of $G$, namely its center, such that $B\cdot B$ contains at least $|B|/p$ cosets of $G^{\prime}$.

For $Q$ a finite quasifield, we similarly define a subset $B\subseteq H_{n}(Q)$ to be a brick if

[TABLE]

where $\underline{X}=X_{1}\times\dots\times X_{n}$ and $\underline{Y}=Y_{1}\times\dots\times Y_{n}$ with non empty-subsets $X_{i},Y_{i},Z\subseteq Q$.

Theorem 1.4**.**

Let $Q$ be a finite quasifield of order $q$. For every $\varepsilon>0$, there exists an $n_{0}=n_{0}(\varepsilon)$ such that if $n\geq n_{0}$, $B\subseteq H_{n}(Q)$ is a brick, and

[TABLE]

then there exists a non trivial subquasigroup $G^{\prime}$ of $H_{n}(Q)$, namely its center $[\underline{0},\underline{0},Q]$ such that $B\cdot B$ contains at least $|B|/q$ cosets of $G^{\prime}$.

Taking $Q=\mathbb{F}_{p}$ gives Theorem 1.1 as a corollary.

2 Preliminaries

2.1 A description of $M_{n}$

We give a description of $M_{n}$ with which it is convenient to work. Define a group $G$ whose elements are triples $[\underline{x},\underline{y},z]$ where $\underline{x}=(x_{1},\dots,x_{n})$, $\underline{y}=(y_{1},\dots,y_{n})$, with $x_{i},y_{i},z\in\mathbb{F}_{p}$ for $1\leq i\leq n$. The group operation in $G$ is given by

[TABLE]

where the function $f:\mathbb{Z}^{n}\times\mathbb{Z}^{n}\rightarrow\mathbb{N}$ is defined by

[TABLE]

where the notation $x\bmod{p}\in\{0,1,\cdots,p-1\}$ means to take the element $x\in\mathbb{F}_{p}$ and consider it as an integer. Concretely, $f$ counts the number of components where (after reducing mod $p$) $y_{i}+y_{i}^{\prime}\geq p$. This is slight abuse of notation, as $\underline{y},\underline{y}^{\prime}\in\mathbb{F}_{p}^{n}$, but is well-defined if we regard them as elements of $\mathbb{Z}^{n}$.

Lemma 2.1**.**

With the operation defined above, $G$ is a group isomorphic to $M_{n}$.

Proof.

We first need to check associativity of the operation. If $f\equiv 0$, we would obtain the Heisenberg group, thus for associativity, after cancellation, it remains to prove

[TABLE]

or equivalently

[TABLE]

The expression

[TABLE]

can be checked to be equal to the left hand side. Comparing it with the right hand side it is enough to formally change the variables $y_{i}$ and $y_{i}^{\prime\prime}$. Both the left and right hand side count the largest multiple of $p$ less than or equal to

[TABLE]

Since $G$ is generated $\{[\underline{e_{i}},0,0],[0,\underline{e_{i}},0],[0,0,1]\}$, we define a homomorphism $\varphi:G\to M_{n}$ by $\varphi\left([\underline{e_{i}},0,0]\right)=a_{i}$, $\varphi\left([0,\underline{e_{i}},0]\right)=b_{i}$, and $\varphi\left([0,0,1]\right)=c$. This map is clearly surjective and it is easy to check that the generators of $G$ satisfy the relations in $M_{n}$. Since $|G|=p^{2n+1}$, $\varphi$ is an isomorphism and $G\cong M_{n}$, as claimed. ∎

With this description, there is a natural way to define a brick in $M_{n}$. A subset $B$ of $M_{n}$ is said to be a brick if

[TABLE]

where $\underline{X}=X_{1}\times\dots\times X_{n}$ and $\underline{Y}=Y_{1}\times\dots\times Y_{n}$ with non empty-subsets $X_{i},Y_{i},Z\subseteq\mathbb{F}_{p}$.

2.2 Tools from spectral graph theory

For a graph $G$ with vertex set $\{v_{1},\dots,v_{n}\}$, the adjacency matrix of $G$ is the matrix with a $1$ in row $i$ and column $j$ if $(v_{i},v_{j})$ is an edge and a [math] otherwise. Since this is a real, symmetric matrix, it has a full set of real eigenvalues. Let $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}$ be the eigenvalues of its adjacency matrix.

If $G$ is a $d$-regular graph, then its adjacency matrix has row sum $d$. In this case, $\lambda_{1}=d$ with the all-one eigenvector $\mathbf{1}$. Let $\mathbf{v}_{i}$ denote the corresponding eigenvector for $\lambda_{i}$. We will make use of the trick that for $i\geq 2$, $\mathbf{v}_{i}\in\mathbf{1}^{\bot}$, so $J\mathbf{v}_{i}=0$ where $J$ is the all-one matrix of size $n\times n$ (see [3] for more background on spectral graph theory).

It is well-known (see [1, Chapter 9] for more details) that if $\lambda_{2}$ is much smaller than the degree $d$, then $G$ has certain random-like properties. A graph is called bipartite if its vertex set can be partitioned into two parts such that all edges have one endpoint in each part. For $G$ be a bipartite graph with partite sets $P_{1}$ and $P_{2}$ and $U\subseteq P_{1}$ and $W\subseteq P_{2}$, let $e(U,W)$ be the number of pairs $(u,w)$ such that $u\in U$, $w\in W$, and $(u,w)$ is an edge of $G$. We recall the following well-known fact (see, for example, [1]).

Lemma 2.2** (Corollary 9.2.5, [1]).**

Let $G=(V,E)$ be $d$-regular bipartite graph on $2n$ vertices with partite sets $P_{1}$ and $P_{2}$. For any two sets $B\subseteq P_{1}$ and $C\subseteq P_{2}$, we have

[TABLE]

2.3 Sum-product graphs

Let $Q$ be a finite quasifield. The sum-product graph $SP_{Q,n}$ is defined as follows. $SP_{Q,n}$ is a bipartite graph with its vertex set partitioned into partite sets $\mathbf{X}$ and $\mathbf{Y}$, where $\mathbf{X}=\mathbf{Y}=Q^{n}\times Q$. Two vertices $U=(\underline{x},z)\in\mathbf{X}$ and $V=(\underline{y},z^{\prime})\in\mathbf{Y}$ are connected by an edge, $(U,V)\in E(SP_{Q,n})$, if and only if $\langle\underline{x},\underline{y}\rangle=z+z^{\prime}$. We need information about the eigenvalues of $SP_{Q,n}$.

Lemma 2.3**.**

If $Q$ is a quasifield of order $q$, then the graph $SP_{Q,n}$ is $q^{n}$ regular and has $\lambda_{2}\leq 2^{1/2}q^{n/2}$.

We provide a proof of Lemma 2.3 for completeness in the appendix, and we note that similar lemmas were proved in [11] and [10].

3 Proof of Theorem 1.2

Lemma 3.1**.**

Let $B\subseteq M_{n}$ be a brick in $M_{n}$ with $B=[\underline{X},\underline{Y},Z]$ where $\underline{X}=X_{1}\times\cdot\times X_{n}$ and $\underline{Y}=Y_{1}\times\dots\times Y_{n}$. For given $\underline{a}=(a_{1},\dots,a_{n}),\underline{b}=(b_{1},\dots,b_{n})\in\mathbb{F}_{p}^{n}$, suppose that

[TABLE]

then we have

[TABLE]

Proof.

Let $X_{i}^{\prime}=X_{i}\cap(a_{i}-X_{i})$, $Y_{i}^{\prime}=Y_{i}\cap(b_{i}-Y_{i}),X^{\prime}=(X_{1}^{\prime},\dots,X_{n}^{\prime})$, and $Y^{\prime}=(Y_{1}^{\prime},\dots,Y_{n}^{\prime})$. We first have

[TABLE]

On the other hand, it follows from the multiplicative rule in $M_{n}$ that for

[TABLE]

To conclude the proof of the lemma, it is enough to prove that

[TABLE]

under the condition $|Z|^{2}|X^{\prime}||Y^{\prime}|>2p^{n+2}$.

To prove this claim, let $\lambda$ be an arbitrary element in $\mathbb{F}_{p}$, we define two sets in the sum-product graph $SP_{\mathbb{F}_{p},n}$, $E\subseteq\mathbf{X}$ and $F\subseteq\mathbf{Y}$ as follows:

[TABLE]

It is clear that $|E|=|Z||X^{\prime}|$ and $|F|=|Z||Y^{\prime}|$. It follows from Lemma 2.2 and Lemma 2.3 that if $|Z|^{2}|X^{\prime}||Y^{\prime}|>2p^{n+2}$, then $e(E,F)>0$. It follows that there exist $\underline{x}\in X^{\prime},\underline{y}\in Y^{\prime}$, and $z,z^{\prime}\in Z$ such that

[TABLE]

Since $\lambda$ is chosen arbitrarily, we have

[TABLE]

Proof of Theorem 1.2.

We follow the method of [9, Theorem 1.3]. First we note that if $|Z|>p/2$, then we have $Z+Z=\mathbb{F}_{p}$. This implies that

[TABLE]

Therefore, $B\cdot B$ contains at least $|B\cdot B|/p\geq|B|/p$ cosets of the subgroup $[\underline{0},\underline{0},\mathbb{F}_{p}]$. Thus, in the rest of the proof, we may assume that $|Z|\leq p/2$.

For $1\leq i\leq n$, we have

[TABLE]

which implies that

[TABLE]

Therefore we obtain

[TABLE]

Let $N$ be the number of pairs $(\underline{a},\underline{b})\in\mathbb{F}_{p}^{n}\times\mathbb{F}_{p}^{n}$ such that

[TABLE]

It follows from Lemma 3.1 that $[\underline{a},\underline{b},\mathbb{F}_{p}]\subseteq B\cdot B$ for such pairs $(\underline{a},\underline{b})$. Then by equation (1)

[TABLE]

and so

[TABLE]

By the assumption of Theorem 1.2, we have

[TABLE]

Thus when $n>1/\epsilon$, we have

[TABLE]

since $|Z|\leq p$.

In other words,

[TABLE]

since $|Z|\leq p/2$. ∎

4 Proof of Theorem 1.4

Lemma 4.1**.**

Let $Q$ be a quasifield of order $q$ and let $[\underline{X},\underline{Y},Z]=B\subseteq H_{n}(Q)$ be a brick. For a given $\underline{a}=(a_{1},\dots,a_{n})$, $\underline{b}=(b_{1},\dots,b_{n})\in Q^{n}$, suppose that

[TABLE]

then we have

[TABLE]

Proof.

The proof is similar to that of Lemma 3.1, so we leave some details to the reader. Let

[TABLE]

and $E\subseteq\mathbf{X}$, $F\subseteq\mathbf{Y}$ in $SP_{Q,n}$ where

[TABLE]

and $\lambda\in Q$ is arbitrary. Then $e(E,F)>0$ which implies that there exist $\underline{x}\in X^{\prime}$, $\underline{y}\in Y^{\prime}$, and $z,z^{\prime}\in Z$ such that

[TABLE]

This implies that

[TABLE]

The rest of the proof of Theorem 1.4 is identical to that of Theorem 1.2. We need only to show that if $Z\subseteq Q$ and $|Z|>|Q|/2$, then $Z+Z=Q$. However, this follows since the additive structure of $Q$ is a group.

Appendix

Proof of Lemma 2.3.

Let $Q$ be a finite quasifield of order $q$ and let $SP_{Q,n}$ be the bipartite graph with partite sets $\mathbf{X}=\mathbf{Y}=Q^{n}\times Q$ where $(x_{1},\dots,x_{n},z_{x})\sim(y_{1},\dots,y_{n},z_{y})$ if and only if

[TABLE]

First we show that $SP_{Q,n}$ is $q^{n}$ regular. Let $(x_{1},\dots,x_{n},z_{x})$ be an arbitrary element of $\mathbf{X}$. Choose $y_{1},\dots,y_{n}\in Q$ arbitrarily. Then there is a unique choice for $z_{y}$ that makes (3) hold, and so the degree of $(x_{1},\dots,x_{n},z_{x})$ is $q^{n}$. A similar argument shows the degree of each vertex in $\mathbf{Y}$ is $q^{n}$.

Next we show that $\lambda_{2}$ is small. Let $M$ be the adjacency matrix for $SP_{Q,n}$ where the first $q^{n+1}$ rows and columns are indexed by $\mathbf{X}$. We can write

[TABLE]

where $N$ is the $q^{n+1}\times q^{n+1}$ matrix whose $(x_{1},\dots,x_{n},x_{z})_{X}\times(y_{1},\dots,y_{n},y_{z})_{Y}$ entry is $1$ if (3) holds and [math] otherwise.

The matrix $M^{2}$ counts the number of walks of length $2$ between vertices. Since $SP_{Q,n}$ is $q^{n}$ regular, the diagonal entries of $M^{2}$ are all $q^{n}$. Since $SP_{Q,n}$ is bipartite, there are no walks of length $2$ from a vertex in $\mathbf{X}$ to a vertex in $\mathbf{Y}$. Now let $x=(x_{1},\dots,x_{n},x_{z})$ and $x^{\prime}=(x_{1}^{\prime},\dots,x_{n}^{\prime},x_{z}^{\prime})$ be two distinct vertices in $\mathbf{X}$. To count the walks of length $2$ between them is equivalent to counting their common neighbors in $\mathbf{Y}$. That is, we must count solutions $(y_{1},\dots,y_{n},z_{y})$ to the system of equations

[TABLE]

and

[TABLE]

Case 1: For $i\leq 1\leq n$ we have $x_{i}=x_{i}^{\prime}$: In this case we must have $x_{z}\not=x_{z}^{\prime}$. Subtracting (4) from (5) shows that the system has no solutions and so $x$ and $x^{\prime}$ have no common neighbors.

Case 2: There is an $i$ such that $x_{i}\not=x_{i}^{\prime}$: Subtracting (5) from (4) gives

[TABLE]

There are $q^{n-1}$ choices for $y_{1},\dots,y_{i-1},y_{i+1},\dots y_{n}$. Since $x_{i}-x_{i}^{\prime}\not=0$, these choices determine $y_{i}$ uniquely, which then determines $y_{z}$ uniquely. Therefore, in this case $x$ and $x^{\prime}$ have exactly $q^{n-1}$ common neighbors.

A similar argument shows that for $y=(y_{1},\dots,y_{n},y_{z})$ and $y^{\prime}=(y_{1}^{\prime},\dots,y_{n}^{\prime},y_{z}^{\prime})$, then either $y$ and $y^{\prime}$ have either no common neighbors or exactly $q^{n-1}$ common neighbors.

Now let $H$ be the graph whose vertex set is $\mathbf{X}\cup\mathbf{Y}$ and two vertices are adjacent if and only if they are either both in $\mathbf{X}$ or both in $\mathbf{Y}$, and they have no common neighbors. For this to occur, we must be in Case 1, and therefore we must have either $x_{z}\not=x_{z}^{\prime}$ or $y_{z}\not=y_{z}^{\prime}$ and all of the other coordinates equal. Therefore, this graph is $q-1$ regular, as for each fixed vertex there are exactly $q-1$ vertices with a different last coordinate and the same entries on the first $n$ coordinates. Let $E$ be the adjacency matrix of $H$ and note that since $H$ is $q-1$ regular, all of the eigenvalues of $E$ are at most $q-1$ in absolute value. Let $J$ be the $q^{n+1}$ by $q^{n+1}$ all ones matrix. By the above case analysis, it follows that

[TABLE]

Now let $v_{2}$ be an eigenvector of $M$ for $\lambda_{2}$. For a set of vertices $Z$ let $\chi_{Z}$ denote the vector which is $1$ if a vertex is in $Z$ and [math] otherwise (ie it is the characteristic vector for $Z$). Note that since $SP_{Q,n}$ is a regular bipartite graph, we have that $\lambda_{1}=q^{n}$ with corresponding eigenvector $\chi_{\mathbf{X}}+\chi_{\mathbf{Y}}$ and $\lambda_{n}=-q^{n}$ with corresponding eigenvector $\chi_{\mathbf{X}}-\chi_{\mathbf{Y}}$. Also note that $v_{2}$ is perpendicular to both of these eigenvectors and therefore is also perpendicular to both $\chi_{\mathbf{X}}$ and $\chi_{\mathbf{Y}}$. This implies that

[TABLE]

Now by (7), we have

[TABLE]

Therefore $q-1-\frac{\lambda_{2}^{2}}{q^{n-1}}$ is an eigenvalue of $E$ and is therefore at most $q-1$ in absolute value, implying that $\lambda_{2}\leq 2^{1/2}q^{n/2}$.

∎