MINIMALLY INTERSECTIVE POLYNOMIALS WITH ARBITRARILY LONG FACTORIZATION
Bhawesh Mishra 111The author was partly supported by the NSF, under grant
DMS-1812028.
Department of Mathematics, The Ohio State University, Columbus, OH USA
<EMAIL_ADDRESS>

Abstract

Given a natural number $n\geq 4$, we show that there exist infinitely many
polynomials $f_{n}(x):=(x^{2}-a_{1})(x^{2}-a_{2})\cdots(x^{2}-a_{n})$ such
that $(i)$ $f_{n}(x)$ has a root modulo every positive integer, $(ii)$
$f_{n}(x)$ has no rational roots, and $(iii)$ every proper divisor of
$f_{n}(x)$ fails to have a root modulo some positive integer. We will call
such polynomials minimally intersective. Our proof also shows that once
$a_{1},a_{2},\ldots,a_{n-1}$ are chosen, the set of natural numbers $a_{n}$
such that the polynomial
$f_{n}(x):=(x^{2}-a_{1})(x^{2}-a_{2})\cdots(x^{2}-a_{n})$ is minimally
intersective has positive asymptotic density in $\mathbb{N}$.

##  1\. Introduction

A set $S\subset\mathbb{Z}$ is called intersective if given any set T
$\subset\mathbb{Z}$ with positive upper density, one has $S$ $\cap$
$(T-T)\not\subseteq\\{0\\}$. Given a $T\subset\mathbb{Z}$, $(T-T)$ is defined
as $\\{t_{1}-t_{2}:t_{1},t_{2}\in T\\}$ and the upper density of $T$ is
defined as

$\overline{d}(T):=\limsup_{n\rightarrow\infty}\frac{|T\cap\\{-n,\ldots,-2,-1,0,1,2,\ldots,n\\}|}{2n+1}.$

A polynomial $f(x_{1},\ldots,x_{m})\in\mathbb{Z}[x_{1},\ldots,x_{m}]$ is said
to be intersective if the set of its values
$\\{f(x_{1},\ldots,x_{m}):x_{1},\ldots,x_{m}\in\mathbb{Z}\\}$ is intersective.
Sárközy and Furstenberg independently and concurrently proved, in [9] and [3]
respectively, that for any subset $T$ of integers with positive upper density,
the set $(T-T)$ contains a perfect square. In other words, they proved that
the polynomial $f(x)=x^{2}$ is intersective.

Kamae and Mendés-France, in [6], showed that a polynomial $f$ of one variable
is intersective if and only if $f(x)\equiv 0\hskip 2.84526pt(\text{mod }m)$ is
solvable for every positive integer $m>1$. A very special case of the
polynomial Szemerédi’s theorem, obtained by Bergelson, Leibman and Lesigne in
[2], generalizes this fact to the polynomials of many variables. Their result
implies that a polynomial
$g(x_{1},\ldots,x_{k})\in\mathbb{Z}[x_{1},\ldots,x_{k}]$ is intersective if
and only if the congruence $g(n_{1},n_{2},\ldots,n_{k})\equiv 0\hskip
2.84526pt(\text{mod }m)$ is solvable for every $m>1$.

Berend and Bilu, in [1], obtained a criterion for any polynomial $f$ of one
variable to be intersective. An implication of their result is that any
polynomial in one variable that is intersective, but has no rational root, has
to be of degree greater than 4. On the other hand, there are single-variable
intersective polynomials of degree greater than $4$ that have no rational
roots. Hyde, Lee and Spearman obtained an infinite family of intersective
polynomials of the form

$\displaystyle h(x)=(x^{3}-n)(x^{2}+3),$

none of which have rational roots [5]. One can easily show that if $p,q$ are
distinct odd primes such that $p\equiv q\equiv 1\hskip 2.84526pt(\text{mod
}4)$ and $\big{(}\frac{p}{q}\big{)}=+1$ then the polynomial

$\displaystyle f(x)=(x^{2}-p)(x^{2}-q)(x^{2}-pq)$

is intersective. Hyde and Spearman generalized this result to the case when
$p$ and $q$ are replaced by square-free integers. Let $c$ and $d$ be square-
free integers not equal to $1$, let $c_{1}=\frac{c}{\text{gcd}(c,d)}$ and
$d_{1}=\frac{d}{\text{gcd}(c,d)}$. Hyde and Spearman obtained necessary and
sufficient conditions for the polynomial

$\displaystyle p(x)=(x^{2}-c)(x^{2}-d)(x^{2}-c_{1}d_{1})$

to be intersective but have no rational root [4]. This result was extended in
[7] by obtaining a necessary and sufficient condition for polynomials of the
form

$\displaystyle f(x)=(x^{2}-a_{1})(x^{2}-a_{2})\cdots(x^{2}-a_{n})$

to be intersective without having a rational root (see Proposition $1$ below).
Here $n\geq 3$ and $a_{1},a_{2},\ldots,a_{n}$ are square-free integers not
equal to $1$.

Note that if $f(x)$ is an intersective polynomial then for every polynomial
$g(x)$, $f(x)g(x)$ is also intersective. Therefore, one could always get more
examples of intersective polynomials by multiplying a given intersective
polynomial $f(x)$ by any polynomial $g(x)$. The topic of this article is to
show existence of, and construct, a particular type of intersective
polynomial, which is defined below.

######  Definition 1.

A polynomial $f(x)\in\mathbb{Z}[x]$ is said to be minimally intersective if it
satisfies the following two conditions:

  1. (1)

$f(x)$ is intersective but $f(x)$ does not have a rational root.

  2. (2)

None of the proper divisors of $f(x)$ is intersective.

Minimally intersective polynomials can be thought of as genuinely new examples
of intersective polynomials because they are not obtained by adjoining factors
to an already intersective polynomials. The result in this article shows that
for every $n\geq 4$, there exist minimally intersective polynomials with $n$
quadratic factors.

Given an integer $n$ and a prime $p$ we will denote the group of quadratic
residues modulo $p$ by $Q_{p}$ and the Legendre symbol of $n$ with respect to
$p$ by $\big{(}\frac{n}{p}\big{)}$. Similarly, $p^{a}\mid\mid n$ $(a\geq 1)$
will denote that $p^{a}$ is the highest power of prime $p$ dividing the
integer $n,(a,b)$ will denote the greatest common divisor of two natural
numbers $a$ and $b$, and rad$(x)$ will denote the square-free part of natural
number $x$. The asymptotic density of a set $A\subset\mathbb{N}$ is defined
as:

$\lim_{n\rightarrow\infty}\frac{|A\cap\\{1,2,\ldots,n\\}|}{n}.$

Let $p$ be an odd prime, $e\geq 1$ and $a\in\mathbb{Z}$ such that $p\nmid a$.
Then $a$ is a square modulo $p^{e}$ if and only if
$\big{(}\frac{a}{p}\big{)}=+1$. This fact is an immediate consequence of
Hensel’s lemma for the polynomial $(x^{2}-a)$. The main result of this article
is the following theorem.

######  Theorem 1.

For every $n\geq 4$, natural numbers $\\{a_{i}\\}_{i=1}^{n}$ can be chosen
such that the polynomial $f_{n}(x):=(x^{2}-a_{1})\cdots(x^{2}-a_{n})$ is
minimally intersective. In fact, once $a_{1},a_{2},\ldots,a_{n-1}$ are chosen,
the set of natural numbers $a_{n}$ for which $f_{n}(x)$ is minimally
intersective, has positive asymptotic density.

We will collect some preliminary results in Section $2$. In Section $3$, we
shall describe a process to find square-free integers $a_{1},\ldots,a_{n}$
that define the polynomials $f_{n}(x)$ for the corresponding $n\geq 4$.
Section $4$ contains proof that the polynomial $f_{n}(x)$ thus obtained is
intersective. The proof that $f_{n}(x)$ is minimally intersective is contained
in Section $5$. Section $6$ contains an explicit example of $f_{4}(x)$ and
$f_{5}(x)$ each.

##  2\. Preliminaries

We will repeatedly utilize the following characterization of intersectivity of
polynomials consisting of quadratic factors that is proved in [7].

######  Proposition 1.

Let $n\geq 3$ and let $a_{1},a_{2},\ldots,a_{n}$ be distinct nonzero square-
free integers, none of which is $1$. Then the polynomial
$f(x)=\prod_{i=1}^{n}(x^{2}-a_{i})$ is intersective if and only if the
following conditions are satisfied.

  1. (1)

There exists $T\subset\\{1,2,\ldots,n\\}$ of odd cardinality such that:

    1. (a)

the product $\prod_{j\in T}a_{j}$ is a perfect square, and

    2. (b)

for every $j\in T$ and for every odd prime $p$ dividing $a_{j}$, there exists
$i\in\\{1,\ldots,n\\}$, $i\neq j$, such that
$\big{(}\frac{a_{i}}{p}\big{)}=+1$ .

  2. (2)

One of the $a_{i}$ is of the form $8m+1$ for some $m\in\mathbb{Z}$ and $m\neq
0$.

We will also use the following classical result about distribution of square-
free integers in an arithmetic progressions that was originally proved in [8].

######  Proposition 2.

Let $a,b\in\mathbb{N}$ such that $(a,b)$ is square-free. Then the density of
the set of square-free natural numbers congruent to $b$ modulo $a$ is
$\frac{6}{\pi^{2}}\prod_{p\mid a}\big{(}1-\frac{1}{p^{2}}\big{)}^{-1}$.

For the sake of brevity, we will present the following elementary fact
(without proof) as a lemma.

######  Lemma 1.

Let $n\geq 3$ and let $a_{1},a_{2},\ldots,a_{n}$ be distinct square-free
nonzero integers, none of which is equal to $1$. Then $\prod_{i=1}^{n}a_{i}$
is a perfect square if and only if for every $j\in\\{1,2,\ldots,n\\}$,
$a_{j}=rad$ $\big{(}\prod_{i=1,i\neq j}^{n}a_{i}\big{)}$.

Now we will state and prove some elementary number-theoretic lemmas that we
shall repeatedly utilize in our proofs.

######  Lemma 2.

Let $p$ be an odd prime and let $a\in\mathbb{Z}$ be a square-free integer. If
$\big{(}\frac{a}{p}\big{)}\neq+1$ then $a$ cannot be a square modulo $p^{2}$.

###### Proof.

If $p\nmid a$ then $\big{(}\frac{a}{p}\big{)}\neq+1$ implies that $a$ cannot
be a square modulo $p^{2}$. On the other hand, if $p\mid a$ then $p\mid\mid a$
because $a$ is square-free.

Assume for the sake of contradiction that $a$ is a square modulo $p^{2}$ ,
i.e., $x^{2}\equiv a\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp^{2})$ for
some $x\in\mathbb{Z}$. Then we have $p^{2}\mid(x^{2}-a)$ , i.e.,
$p\mid(x^{2}-a)$. Since $p\mid a$ and $p\mid(x^{2}-a)$ we have that $p\mid
x^{2}$ implying $p^{2}\mid x^{2}$. However, $p^{2}\mid(x^{2}-a)$ and
$p^{2}\mid x^{2}$ gives that $p^{2}\mid a$, contradicting that $a$ is square-
free. Therefore, $a$ cannot be a square modulo $p^{2}$. ∎

######  Lemma 3.

Let $k,m\in\mathbb{N}$, let $p$ be a prime and let
$f(x)=\prod_{i=1}^{m}(x^{2}-a_{i})\in\mathbb{Z}[x]$. If $(x^{2}-a_{i})\equiv
0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp^{k})$ is not solvable for any
$1\leq i\leq m$ then $f(x)\equiv 0\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp^{km})$ is not solvable.

###### Proof.

For the sake of contradiction, assume that $f(x)\equiv 0\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp^{km})$ is solvable for some
$x\in\mathbb{Z}$ , i.e., $p^{km}\mid(x^{2}-a_{1})\cdots(x^{2}-a_{m})$. Then we
must have $p^{k}\mid(x^{2}-a_{j})$, for some $j\in\\{1,2,\ldots,m\\}$.

However $p^{k}\mid(x^{2}-a_{j})$ implies that $(x^{2}-a_{j})\equiv 0\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp^{k})$ is solvable, a contradiction to
the fact that $(x^{2}-a_{i})\equiv 0\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp^{k})$ is not solvable for any $i$. Therefore, we have the result. ∎

######  Lemma 4.

Let $k\geq 3$ and let $a_{1},\ldots,a_{k}$ be distinct, nonzero square-free
integers. Suppose that for each $1\leq m\leq k$ and for every
$T\subset\\{1,2,\ldots,m\\}$, $a_{m}>\text{rad }\big{(}\prod_{j\in T,j\neq
m}a_{j}\big{)}$. Then for any subset $S$ $\subseteq\\{1,2,\ldots,k\\}$,
$\prod_{j\in S}a_{j}$ is not a perfect square.

###### Proof.

For the sake of contradiction, assume that there is an
$S\subseteq\\{1,2,\ldots,k\\}$ such that $\prod_{j\in S}a_{j}$ is a perfect
square. Then using Lemma $1$, we have that

$a_{j_{0}}=\text{rad }\big{(}\prod_{j\in S,j\neq j_{0}}a_{j}\big{)},$

where $j_{0}=$ max $S$. This is a contradiction to our assumption for
$m=j_{0}$ and $T=S\subset\\{1,2,\ldots,j_{0}\\}$. Hence we have the result. ∎

##  3\. Finding $a_{1},a_{2},\ldots,a_{n}$ such that
$f_{n}(x)=\prod_{i=1}^{n}(x^{2}-a_{i})$

In this section, we will show the existence of natural numbers
$a_{1},a_{2},\ldots,a_{n}$ that will define the polynomial
$f_{n}(x)=(x^{2}-a_{1})(x^{2}-a_{2})\cdots(x^{2}-a_{n})$. Each of the integers
$a_{1},\ldots,a_{n}$ will be square-free and not equal to $1$. They can be
chosen in accordance with the following steps.

  1. (1)

Pick distinct odd primes $p_{1}$ and $p_{2}$. For each $i=1,2$ pick nonzero
elements $b_{i},c_{i}\in\big{(}\mathbb{Z}/p_{i}\mathbb{Z}\big{)}^{*}$ such
that $c_{i}\in Q_{p_{i}}$ $b_{i}\not\in Q_{p_{i}}$. Set $a_{1}=p_{1}p_{2}$.

We can choose $a_{1}$ as any square-free odd natural number that has at least
two odd prime factors and choose $p_{1},p_{2}$ as two odd primes dividing
$a_{1}$. For the sake of brevity, we choose $a_{1}$ to be a product of two odd
primes.

  2. (2)

Pick square-free $a_{2}\in\mathbb{N}$ such that $a_{2}>a_{1}$ and $a_{2}\equiv
b_{j}\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{j})$ for $j=1,2$. Any
integer $a_{2}$ satisfying $a_{2}\equiv b_{j}\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{j})$ for $j=1,2$ above is unique modulo $p_{1}p_{2}$, as a
consequence of the Chinese remainder theorem.

Since $p_{1}\neq p_{2}$, $p_{1}p_{2}$ is square-free and hence infinitely many
square-free $a_{2}$ exist, by Proposition $2$. We pick a square-free $a_{2}$
greater than $a_{1}$. Also note that since $(b_{j},p_{j})=1$ for every $j=1,2$
we have that $(a_{2},p_{j})=1$ for $j=1,2$.

  3. (3)

For any $i\leq n-3$, after choosing $a_{1},\ldots,a_{i-1}$, choose a square-
free integer $a_{i}$ such that:

     * •

$a_{i}\equiv b_{j}\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{j})$ for
$j=1,2$ and

     * •

for every $A\subset\\{1,\ldots,i-1\\}$, $a_{i}$ is greater than the rad
$\big{(}\prod_{j\in A}a_{j}\big{)}$.

Exactly as in step $2$, a square free $a_{i}$ satisfying the above exists as a
consequence of Proposition $2$ and $(a_{i},p_{j})=1$ for $j=1,2$.

  4. (4)

Choose a square-free natural number $a_{n-2}$ that satisfies the following
requirements.

     * •

$a_{n-2}\equiv\begin{cases}b_{1}\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{1});&\text{if $n$ is even}\\\ c_{1}\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp_{1});&\text{if $n$ is
odd}\par\end{cases}$

     * •

$a_{n-2}\equiv b_{2}\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{2})$

     * •

$a_{n-2}$ is greater than rad $\big{(}\prod_{j\in T}a_{j}\big{)}$, for any
$T\subset\\{1,\ldots,n-3\\}$.

Such a square-free $a_{n-2}$ exists, again due to Proposition $2$. Similarly,
we also have $(a_{n-2},p_{j})=1$ for $j=1,2$.

  5. (5)

Define square-free natural number $a_{n-1}$ as:

$a_{n-1}=\begin{cases}\text{rad }\big{(}\prod_{i=1}^{n-2}a_{j}\big{)};&\text{
if $n$ is even}\\\ \text{rad }\big{(}\prod_{i=1}^{n-3}a_{j}\big{)};&\text{ if
$n$ is odd.}\par\end{cases}$

  6. (6)

Choose all the odd primes $p_{1},\ldots,p_{M}$ dividing any of
$a_{1},\ldots,a_{n-1}$ and pick $c_{j}\in Q_{p_{j}}$ for every $1\leq j\leq
M$. Now, pick a square-free natural number $a_{n}$ that satisfies the
following.

     * •

$a_{n}\equiv c_{j}\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{j})$ for any
$2\leq j\leq M$

     * •

$a_{n}\equiv\begin{cases}c_{1}\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{1});&\text{ if $n$ is even}\\\ b_{1}\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp_{1});&\text{ if $n$ is
odd}\par\end{cases}$

     * •

$a_{n}\equiv 1\hskip 2.84526pt(\text{mod}\hskip 2.84526pt8)$

     * •

For any subset $S\subset\\{1,2,\ldots,n-1\\}$, $a_{n}>\text{ rad
}\big{(}\prod_{j\in S}a_{j}\big{)}$.

Any integer $a_{n_{0}}$ that satisfies above conditions is unique modulo
$(8p_{1}\cdots p_{M})$, as a consequence of the Chinese remainder theorem. In
other words, if $a_{n_{0}}$ satisfies above congruences then any integer in
the arithmetic progression $\big{(}8p_{1}\cdots
p_{M}\big{)}\mathbb{N}+a_{n_{0}}\big{)}$ also satisfies those congruences.

Since $a_{n_{0}}$ is odd and $(c_{i},p_{i})=1$ for $i=1,2,\ldots,M$,
$\big{(}a_{n_{0}},8p_{1}\cdots p_{M}\big{)}$ is square-free. Hence the set of
square-free natural numbers $a_{n}$ that satisfy above congruences is of
positive density, as a consequence of Proposition $2$. We choose a square-free
$a_{n}$ such that for any $S\subset\\{1,2,\ldots,n-1\\}$:

$\displaystyle a_{n}>\text{rad }\big{(}\prod_{j\in S}a_{j}\big{)}$

  7. (7)

Define $f_{n}(x):=(x^{2}-a_{1})(x^{2}-a_{2})\cdots(x^{2}-a_{n})$.

##  4\. Proof that $f_{n}(x)$ is Intersective

In this section, we shall prove that $f_{n}(x)$ has roots modulo every
integer, by showing that $f_{n}(x)$ satisfies the conditions in Proposition
$1$.

  1. (1)

Condition $1(a)$ of Proposition $1$ holds for $f_{n}(x)$ because $\prod_{j\in
T}a_{j}$ is a perfect square, where

$T=\begin{cases}\\{1,2,\ldots,n-2,n-1\\}&\text{ ; if $n$ is even}\\\
\\{1,2,\ldots,n-4,n-3,n-1\\}&\text{ ; if $n$ is odd.}\par\end{cases}$

This follows from Step $5$ of Section $3$ and Lemma $1$.

  2. (2)

Let $j\in T$ and $p$ be any odd prime dividing $a_{j}$, then from the step $6$
of Section $3$ it follows that $p=p_{j}$ for some $1\leq j\leq M$. It also
follows from steps $4$ and $6$ of Section $3$ that:

$\begin{cases}\big{(}\frac{a_{n}}{p}\big{)}=+1&\text{if $j\neq 1$ or $n$ is
even}\\\ \big{(}\frac{a_{n-2}}{p}\big{)}=+1&\text{if $j=1$ and $n$ is
odd.}\par\end{cases}$

Therefore the condition $1(b)$ of Proposition $1$ is also satisfied for
$f_{n}(x)$.

  3. (3)

The condition $2$ of the Proposition $1$ is satisfied for $f_{n}(x)$ because
$a_{n}$ is chosen to be square-free and equivalent to $1$ modulo $8$ (in Step
$6$ of Section $3$).

##  5\. Proof that $f_{n}(x)$ is Minimally Intersective

In this section, we shall prove that if we remove any quadratic factors from
$f_{n}(x)$, the resulting polynomial $g_{n}(x)$ will fail to be intersective.
To show that $g_{n}(x)$ is not intersective, we will show that $g_{n}(x)$
fails to satisfy the necessary conditions in Proposition $1$. We will separate
the proofs into parts according to the quadratic factor that is being removed
from $f_{n}(x)$.

  * •

Removing $(x^{2}-a_{r})$ for any $1\leq r\leq(n-3)$

In Section $3$, we chose the square-free integers
$a_{1},a_{2}\ldots,a_{r-1},a_{r+1},\ldots,a_{n-2},a_{n}$ such that they
satisfy the assumption of Lemma $4$. Therefore, for any subset
$S\subset\\{1,2,\ldots,r-1,r+1,\ldots,n-2,n\\}$, $\prod_{j\in S}a_{j}$ cannot
a perfect square.

Hence, if there exists a set $S\subset\\{1,2,\ldots,r-1,r+1,\ldots,n\\}$ such
that $\prod_{j\in S}a_{j}$ is a perfect square, then $(n-1)\in S$. If $n\in
S$, then by Lemma $1$, we must have that $a_{n}=\text{rad }\big{(}\prod_{j\in
S,j\neq n}a_{j}\big{)}$, which is a contradiction to the way $a_{n}$ was
chosen in the step $6$ of Section $3$.

Therefore $n-1=$ max $S$ and by Lemma $1$ we have that

$a_{n-1}=\text{ rad }\big{(}\prod_{j\in S,j\neq(n-1)}a_{j}\big{)}.$

This, along with Step $5$ of Section $3$ implies

$\text{ rad }\big{(}\prod_{j\in
S,j\neq(n-1)}a_{j}\big{)}=\begin{cases}\text{rad
}\big{(}\prod_{i=1}^{n-2}a_{j}\big{)}&\text{ ; if $n$ is even}\\\ \text{rad
}\big{(}\prod_{i=1}^{n-3}a_{j}\big{)}&\text{ ; if $n$ is odd.}\par\end{cases}$

Hence we obtain that

$\begin{cases}\prod_{j\in S,j\neq(n-1)}a_{j}\times\prod_{i=1}^{n-2}a_{j}\text{
is a perfect square}&\text{ ; if $n$ is even}\\\ \prod_{j\in
S,j\neq(n-1)}a_{j}\times\prod_{i=1}^{n-3}a_{j}\text{ is a perfect
square}&\text{ ; if $n$ is odd.}\par\end{cases}$

Since $r\not\in S$ but $r\in\\{1,2,\ldots,n-3\\}$, the two factors above are
not equal, regardless of whether $n$ is odd or even. Therefore, we could
disregard all $a_{j}$ appearing in both the multiplicands above.

After disregarding the common $a_{j}$ appearing in both the multiplicands, we
are still left with a product of distinct $a_{j}$ that is a perfect square.
Let $r_{0}$ be the largest such that $a_{r_{0}}$ remaining in this product.
Then using Lemma $1$ we have that

$a_{r_{0}}=\text{ rad }\big{(}a_{j_{1}}\times\cdots\times a_{j_{l}}\big{)},$

where $j_{1},\ldots,j_{l}\in\\{1,2,\ldots,r_{0}-1\\}$. Since $a_{r_{0}}$ was
chosen from product of $a_{j}$ with $j\leq(n-2)$, we have that
$r_{0}\leq(n-2)$. This is a contradiction to how $a_{1},a_{2},\ldots,a_{n-2}$
were chosen the steps $1,2$, $3$ and $4$ of Section $3$. Specifically, for
$i\leq n-2$, we had chosen $a_{i}$ to be greater than square-free part of any
sub-product of $a_{1},\ldots,a_{i-1}$.

Therefore if we remove $(x^{2}-a_{r})$ for any $1\leq r\leq(n-3)$ then for the
resulting polynomial $g_{n}(x):=\frac{f_{n}(x)}{(x^{2}-a_{r})}$, the condition
$1(a)$ of Proposition $1$ will not be satisfied. Hence, $g_{n}(x)$ is not
intersective.

  * •

Removing $(x^{2}-a_{n-2})$ from $f_{n}(x)$

If $n$ is even, then removing $(x^{2}-a_{n-2})$ from $f_{n}(x)$ will again
result in a polynomial $g_{n}(x):=\frac{f_{n}(x)}{(x^{2}-a_{n-2})}$ not
satisfying the condition $1(a)$ of Proposition $1$. The proof of this is
exactly analogous to the previous case of removing $(x^{2}-a_{r})$ for $1\leq
r\leq(n-3)$. Therefore, we assume that $n$ is odd.

    * –

Since $\big{(}\frac{a_{1}}{p_{1}}\big{)}=0\neq+1$, $(x^{2}-a_{1})\equiv
0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{1}^{2})$ is not solvable. This
follows from Lemma $2$.

    * –

Since for any $2\leq i\leq n-3$ or $i=n$
$\big{(}\frac{a_{i}}{p_{1}}\big{)}=-1\neq+1$ from step $2,3$ and $6$ of
Section $3$, $(x^{2}-a_{i})\equiv 0\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{1}^{2})$ is not solvable . This again follows from Lemma $2$.

    * –

$(x^{2}-a_{n-1})\equiv 0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{1}^{2})$
is not solvable because $\big{(}\frac{a_{n-1}}{p_{1}}\big{)}=0\neq+1$. This
follows from Lemma $2$ again.

So for the resulting polynomial

$\displaystyle
g_{n}(x)=(x^{2}-a_{1})\cdots(x^{2}-a_{n-3})(x^{2}-a_{n-1})(x^{2}-a_{n})$

the congruence

$\displaystyle g_{n}(x)\equiv 0\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{1}^{2(n-1)})$

is not solvable. This last assertion follows from Lemma $3$ for $k=2$,
$m=(n-1)$ and $p=p_{1}$. Hence, $g_{n}(x)$ is not intersective.

  * •

Removing $(x^{2}-a_{n-1})$ from $f_{n}(x)$

In this case, we note that $a_{1},\ldots,a_{n-2},a_{n}$ satisfies the
hypothesis of Lemma $4$ and hence for every subset
$S\subset\\{1,2,\ldots,n-2,n\\}$, the product $\prod_{j\in S}a_{j}$ is not a
perfect square.

Therefore the polynomial $g_{n}(x):=\frac{f_{n}(x)}{(x^{2}-a_{n-1})}$ does not
satisfy the condition $1(a)$ of Proposition $1$ and hence is not intersective.

  * •

Removing $(x^{2}-a_{n})$ from $f_{n}(x)$

If $n$ is even, then note the following.

    * –

Since $\big{(}\frac{a_{1}}{p_{1}}\big{)}=0\neq+1$, $(x^{2}-a_{1})\equiv
0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{1}^{2})$ is not solvable. This
follows from Lemma $2$.

    * –

Since for any $2\leq i\leq n-2$ $\big{(}\frac{a_{1}}{p_{1}}\big{)}=-1\neq+1$
from step $3$ and $4$ of Section $3$, $(x^{2}-a_{i})\equiv 0\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp_{1}^{2})$ is not solvable . This follows
again from Lemma $2$.

    * –

$(x^{2}-a_{n-1})\equiv 0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{1}^{2})$
is not solvable because $\big{(}\frac{a_{n-1}}{p_{1}}\big{)}=0\neq+1$. This
follows again from Lemma $2$.

Therefore, for the resulting polynomial
$g_{n}(x)=(x^{2}-a_{1})\cdots(x^{2}-a_{n-2})(x^{2}-a_{n-1})$ we have that the
congruence

$\displaystyle g_{n}(x)\equiv 0\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{1}^{2(n-1)})$

is not solvable This last assertion follows from Lemma $3$ for $k=2$,
$m=(n-1)$ and $p=p_{1}$; hence, $g_{n}(x)$ is not intersective.

If $n$ is odd, then we have the following implications.

    * –

Since $\big{(}\frac{a_{1}}{p_{2}}\big{)}=0\neq+1$, $(x^{2}-a_{1})\equiv
0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{2}^{2})$ is not solvable. This
follows from Lemma $2$.

    * –

Since for any $2\leq i\leq n-2$ $\big{(}\frac{a_{1}}{p_{2}}\big{)}=-1\neq+1$
from step $3$ and $4$ of Section $3$, $(x^{2}-a_{i})\equiv 0\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp_{2}^{2})$ is not solvable. This follows
again from Lemma $2$.

    * –

$(x^{2}-a_{n-1})\equiv 0\hskip 2.84526pt(\text{mod}\hskip 2.84526ptp_{2}^{2})$
is not solvable because $\big{(}\frac{a_{n-1}}{p_{2}}\big{)}=0\neq+1$. This
follows again from Lemma $2$.

Therefore, when $n$ is odd, for the resulting polynomial

$\displaystyle g_{n}(x)=(x^{2}-a_{1})\cdots(x^{2}-a_{n-2})(x^{2}-a_{n-1})$

we have that the congruence

$\displaystyle g_{n}(x)\equiv 0\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{2}^{2(n-1)})$

is not solvable. This last assertion again follows from Lemma $3$ for $k=2$,
$m=(n-1)$ and $p=p_{2}$; hence $g_{n}(x)$ is not intersective.

##  6\. Some Examples

We shall construct an explicit example of minimally intersective $f_{4}(x)$
and then another of a minimally intersective $f_{5}(x)$.

###  6.1. An Example of Minimally Intersective $f_{4}(x)$

  1. (1)

We pick $p_{1}=$3 and $p_{2}=5$. And, we pick $c_{1}=1\in Q_{3}$,
$b_{1}=2\not\in Q_{3}$, $c_{2}=1\in Q_{5}$ and $b_{2}=2\not\in Q_{5}$. We
define $a_{1}=p_{1}p_{2}=15$.

  2. (2)

Now we pick a square-free integer $a_{2}>15$ such that $a_{2}\equiv 2\hskip
2.84526pt(\text{mod}\hskip 2.84526pt3)$ and $a_{2}\equiv 2\hskip
2.84526pt(\text{mod}\hskip 2.84526pt5)$. We pick $a_{2}=17$.

  3. (3)

Since $n=4$ is even and $n-1=3$, we take $a_{3}=$ rad $(a_{1}\times a_{2})$ =
rad $(15\times 17)=255$.

  4. (4)

Now we take all the primes $p_{1}=3,p_{2}=5,p_{3}=17$ that divides any one of
the $a_{1},a_{2},a_{3}$. Then for every $1\leq j\leq 3$, we take $c_{j}\in
Q_{p_{j}}$. Here we take $c_{1}=1$, $c_{2}=1$ and $c_{3}=2$.

Then we solve for a square-free $a_{4}$ such that $a_{4}>\text{rad
}(\prod_{J}a_{j})$ for every $J\subseteq\\{1,2,3\\}$, $a_{4}\equiv c_{j}\hskip
2.84526pt(\text{mod}\hskip 2.84526ptp_{j})$ for every $j=1,2,3$ and
$a_{4}\equiv 1\hskip 2.84526pt(\text{mod}\hskip 2.84526pt8)$.

By Proposition $2$, infinitely many such square-free $a_{4}$ exists. We choose
$a_{4}=2161$, which is a prime and hence square-free. Therefore,

$\displaystyle f_{4}(x)=(x^{2}-15)(x^{2}-17)(x^{2}-255)(x^{2}-2161)$

is minimally intersective.

###  6.2. An Example of Minimally Intersective $f_{5}(x)$

  1. (1)

As in $6.1$, take $p_{1}=3$, $p_{2}=5$, $c_{1}=1\in Q_{3}$, $c_{2}=1\in
Q_{5}$, $b_{1}=2\not\in Q_{3}$, $b_{2}=2\not\in Q_{5}$, $a_{1}=15$ and
$a_{2}=17$.

  2. (2)

Pick a square-free integer $a_{3}>15\times 17=255$ such that $a_{2}\equiv
2\hskip 2.84526pt(\text{mod}\hskip 2.84526pt3)$ and $a_{2}\equiv 2\hskip
2.84526pt(\text{mod}\hskip 2.84526pt5)$. We choose $a_{3}=557$, which is a
prime and hence square-free.

  3. (3)

Define $a_{4}$ to be rad $(a_{1}\times a_{2})=15\times 17=255$ and hence
$a_{4}=255$.

  4. (4)

Pick all the odd primes $p_{1}=3,p_{2}=5,p_{3}=17,p_{4}=557$ that divide any
one of the $a_{1},a_{2},a_{3},a_{4}$. Then take $c_{1}=1$, $c_{2}=1$,
$c_{3}=2$ and $c_{4}=6$ which are in $Q_{p_{j}}$ for $j=1,2,3,4$ respectively.

Now choose a square-free $a_{5}$ such that $a_{5}>\prod_{J}a_{j}$ for all
$J\subset\\{1,2,3,4\\}$, $a_{5}\equiv c_{j}\hskip 2.84526pt(\text{mod}\hskip
2.84526ptp_{j})$ for $j=1,2,3,4$ and $a_{5}\equiv 1\hskip
2.84526pt(\text{mod}\hskip 2.84526pt8)$.

Any such $a_{5}$ has to be of the form $587641+142035k$ for some
$k\in\mathbb{Z}$. We take $a_{5}=587641+142035(2)=871711$, which is square-
free since its prime-factorization is $29\times 30059$. Therefore,

$\displaystyle
f_{5}(x)=(x^{2}-15)(x^{2}-17)(x^{2}-557)(x^{2}-255)(x^{2}-871711)$

is minimally intersective.

## References

  * [1] D. Berend and Y. Bilu, Polynomials with roots modulo every integer, Proc. Amer. Math. Soc. 124 (1996), 1663–1671.
  * [2] V. Bergelson, A. Leibman and E. Lesigne, Intersective polynomials and polynomial Szemerédi theorem, Adv. Math. 219 (2008), 369–388.
  * [3] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d’Analyse Math. 71 (1977), 204–256.
  * [4] A. M. Hyde and B. K. Spearman, Products of quadratic polynomials with roots modulo any integer, Int. Math. Forum. 8 (2013), 1225–1231.
  * [5] A. M. Hyde, D. P. Lee and B. K. Spearman, Polynomials $(x^{3}-n)(x^{2}+3)$ solvable modulo any integer, Amer. Math. Monthly. 121 (2014), 355–358.
  * [6] T. Kamae and M. Mendés-France, Van der Corput’s difference theorem, Israel. J. Math. 31 (1978), 335–342.
  * [7] B. Mishra, Polynomials consisting of quadratic factors with roots modulo any positive integer, To Appear in Amer. Math. Monthly (Accepted 02.24.2021), https://arxiv.org/abs/2102.08379v1.
  * [8] K. Prachar, Uber die kleinste quadratfreie einer arithmetischen reihe, Monatsh. Math.  62 (1958), 173–176.
  * [9] A. Sárközy, On difference sets of sequences of integers, Acta. Math. Acad. Sci. Hungar. (1978), 125–149.


# Disadvantaged Communities Have Lower Access to Urban Infrastructure

Leonardo Nicoletti1∗, Mikhail Sirenko1, Trivik Verma1

1Faculty of Technology, Policy and Management,
Delft University of Technology, 2628BX Delft, The Netherlands

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
> Disparity in spatial accessibility is strongly associated with growing
> inequalities among urban communities. Since improving levels of
> accessibility for certain communities can provide them with upward social
> mobility and address social exclusion and inequalities in cities, it is
> important to understand the nature and distribution of spatial accessibility
> among urban communities. To support decision-makers in achieving inclusion
> and fairness in policy interventions in cities, we present an open-source
> and data-driven framework to understand the spatial nature of accessibility
> to infrastructure among the different demographics. We find that
> accessibility to a wide range of infrastructure in any city ($54$ cities)
> converges to a Zipf’s law, suggesting that inequalities also appear
> proportional to growth processes in these cities. Then, assessing spatial
> inequalities among the socioeconomically clustered urban profiles for $10$
> of those cities, we find urban communities are distinctly segregated along
> social and spatial lines. We find low accessibility scores for populations
> who have a larger share of minorities, earn less, and have a relatively
> lower number of individuals with a university degree. These findings suggest
> that the reproducible framework we propose may be instrumental in
> understanding processes leading to spatial inequalities and in supporting
> cities to devise targeted measures for addressing inequalities for certain
> underprivileged communities.

## Introduction

Cities are places of consumption that provide a variety of amenities and
services to their citizens (?). Cities with high-capital investments and
technological dominance have become attractive for migrants who are highly
skilled, well-educated, and well-paid (?, ?). As Nijman et al. (?) argue in a
comprehensive review of urban inequalities; national and regional level
policies around globalisation, migration and free-market investments have
catapulted the urban economy but the benefits have not accrued equitably
across urban populations, and as cities grow, the benefits of living in them
are increasingly unequally distributed (?). Akin to the notion that to survive
a decline, cities must make attractive amenities accessible for increasingly
rich workers (?), the responsibility111Sustainable Development Goal 10
(Reduced Inequalities) suggests that there is a mutual understanding among
world governments to tackle inequalities as a global challenge. (?) of
_equitably_ addressing urban inequalities also lies with municipal authorities
(?).

Over time, the dynamics of gentrification, the housing market, residential
segregation and de jure practices like exclusionary zoning laws (?, ?) have
pushed low-income minority groups to live in deteriorated urban areas
receiving a minor share of public investments (?, ?). Such processes have
resulted in the concentration of urban poverty and the creation of strong
inequalities among urban communities (?, ?, ?, ?, ?). Through sociological
investigations and observations, scholars have illustrated how urban
inequalities are related to interlinked social factors across spatial and
temporal scales (?). In general, depending on their socioeconomic status (?),
some communities benefit from access to a wide range of urban services such as
mobility, health, education and community space, while others are segregated
in neighbourhoods where they struggle to access resources within the city (?,
?, ?, ?, ?). People living in areas with a poor concentration of accessible
services and amenities tend to snowball into low levels of education, poor
physical and mental health, disproportionate job opportunities and social
exclusion (?, ?, ?, ?), thus lacking upward mobility.

As Eric Klinenberg posits in their book that building better social
infrastructure will result in more resilient and less unequal societies, it is
an account of how urban infrastructure can facilitate shared spaces through
which civic life can be rebuilt and improved (?) for vastly underprivileged
communities. Fortunately, globally, cities have implemented policies aimed at
improving the quality of urban infrastructure, and at reducing inequalities in
accessibility among urban communities (?, ?, ?, ?, ?, ?). In the United States
and Canada alone, more than $400$ municipalities have implemented policies to
encourage the development of “complete street” (?). Such policies aim to
create socially inclusive communities with improved access to services, shops
and recreation, healthy and active lifestyles, more walking and less driving,
attractive public spaces, and improved economic vitality (?).

Even though nuanced variations exist (?, ?, ?), more formally, _Accessibility_
refers to the ease with which residents within a city can reach amenities or
opportunities (destinations, social interactions, jobs) (?). It is a general
measure adopted by planners to understand how land-use and transport systems
shape the quality of life of residents in a city and identify where equitable
developments can improve the life of marginalised communities (?). Studies
that attempt to inform policy by quantitatively highlighting urban
inequalities in cities usually measure variability in access for residents to
specific spatial factors, such as access to transit, health or green space
infrastructure, and by specific demographic attributes, such as income or
ethnicity (?, ?, ?). These findings are especially useful for decision-makers
to tailor policies toward their specific communities (?). However,
inequalities among communities are shaped differently depending on the
diversity in distribution of resources and compounded by several other spatial
and economic factors (?), and adopting utilitarian (or case-specific)
approaches may prevent in addressing the causes of inequalities in urban
regions (?). Therefore, systematically understanding the variability in
spatial distribution of access and the associated demographic distribution is
instrumental in designing targeted and equitable policies for addressing urban
inequalities.

Using open-source data, we present a framework to understand and evaluate the
differences in access to urban infrastructure afforded to various demographics
within cities around the world. By taking into consideration more than $50$
types of urban amenities, we quantify accessibility to different categories of
basic services for $54$ cities, globally. Despite the geo-political
differences that exist across these cities, we find that accessibility in all
cities is driven by a universal log-normal distribution. We infer the
existence of this law by comparing the distribution of accessibility to the
spread of population density in those regions. By taking into consideration
several categories of socioeconomic attributes, we characterize city-specific
urban communities and assess inequalities in accessibility in $10$ of those
cities (due to data-availability). Our findings reveal the existence of
structural inequalities for similar archetypes of urban communities across all
North American cities within our data. This approach supports us in
investigating the nature of inequalities in access and its relationship with
different socioeconomic aspects of urban communities, thus providing evidence
to derive tailored policies in addressing inequalities in an equitable manner.

Our goal is twofold: First, we investigate the nature of spatial factors that
relate to accessibility in any given city, and in turn urban inequalities.
Then, because of its modular nature, the framework can be adapted to the local
values of decision-makers with regards to accessibility, and provide support
in identifying region-specific social groups suffering from inequalities in
access within their regions.

## Results

There is more to the scale-free nature of accessibility 222Subtitle inspired
by (?)

To examine how accessible urban infrastructure is for residents in a city, we
first define a measurement of _accessibility_ $A$ (see Section Methods for
details) that quantifies the ease with which residents within a city can reach
amenities (?, ?) (for e.g. Mobility, Active living, Entertainment, Food
Choices, Community Space, Education, or Health and Well Being) by walking or
by using a form of _active_ transportation (e.g. bicycle or roller-blades)
(?). Active accessibility measures (?) play a central role in shaping
accessibility policy worldwide (?, ?) by demonstrating how well urban dwellers
have access to basic services in the immediate proximity of their place of
residence. We make the choice of defining accessibility as active for two
reasons: First, there are many cities that do not have any reasonable forms of
public transportation and presuppose participation in society through private
modes of transportation like cars - using a more general definition of
accessibility will only facilitate a comparison among limited car-dependent
urban regions. Second, our work is responding to policy efforts in promoting
more active accessibility in cities around the world (?).

Regardless of how each city has evolved, the accessibility score $A\in[0,1]$
for the spatial units (see Section Methods for details) of any given city is
either normally distributed (see Figure 1A) or a mixture of normal
distributions (see Supplementary Information (SI) for a detailed analysis of
the distributions). As $A$ is defined using a logarithm, a log-normal
relationship governing the distribution of accessibility across the world
could be explained following the economic and spatial integration of
settlements into a larger urban area (?): Access (as a resource that is
developing with the growth of an urban region) may have evolved log-normally
as a result of the overall effects of factors such as economic, technological
or demographic changes (?) (also see Ref. (?) for justification of using a
log-normal distribution). Though, accessibility score $A_{i}$ of a given
spatial unit $i$ across a city is inversely proportional to the natural log of
its rank in accessibility (see Figure 1B) following
$A_{i}=-0.3+1.25(ln(A_{i}^{r})$ where $A_{i}^{r}$ is the rank of the spatial
unit $i$ in the distribution of a specific city, there is a cut-off to the
distribution implying that certain regions may not be economically well
integrated (?), depending on the sprawl size of the city (?).

Figure 1: (A) Statistical distribution of accessibility for Chicago, Montreal,
San Francisco, Los Angeles, New York City, Philadelphia, Vancouver, Miami,
Seattle and Houston, showing log-normal behaviour. (B) The scale free
relationship between a spatial unit’s accessibility A and the rank of its
accessibility ($A_{i}=-0.3+1.25(ln(A_{i}^{r})$). The dots represent equal
sized grid-cells within and across all cities.

Since accessibility is only defined as a measure of infrastructural provision,
decision-makers may argue that regions with lower populations may not
necessitate the same investment of resources in providing better levels of
accessibility compared to urban centres. To ground the discussion of
accessibility in urban policy, we measure the cumulative size of population
that has accessibility $A$ lower than $a$ in a city using a separate Global
Human Settlement Layer (GHSL) data set of population residing in similar sized
spatial units (?, ?) (see Section Methods for details about GHSL population
grids). Figure 2A shows that, in many North American cities, while there is a
large number of spatial units with high accessibility scores (for example,
$A>0.3$), there is a large proportion of the population with low accessibility
scores (i.e. $A<0.3$), evident from the difference in area between the density
curves. In Houston and Seattle, for example, around $90\%$ of the population
resides in spatial units that have a score $A$ of less than $0.3$. In those
cities, accessibility only benefits a few residents. Figure 2B presents this
distribution for all $54$ cities within our study. It shows that, in cities
where curves are heavily shifted to the right and which start to resemble the
sigmoid curve $S(x)$, $S(x)=\frac{1}{1+e^{-k(x-0.5)}}$ where $k=10$), there is
a larger share of people that benefits from higher accessibility scores, and
the share of people with lower accessibility scores is smaller. Cities like
Zurich, for example, come close, but no city resembles the sigmoid function.
In this context, $S(x)$ can be used as a benchmark for accessibility within
cities, a hypothetical scenario where $50\%$ of the population has at least an
accessibility score $A$ of $0.5$. See SI for more details on descriptive
statistics of accessibility for all cities.

Through this analysis, we are not attempting to resolve the regionalisation
debate (?), wherein, depending on the number of spatial units taken into
account, the distribution of $A$ may transform. Our data represents
administrative units of each city, which may differ from economically
functional units, especially for smaller cities (?) (see Section Methods for
data description). To partly address this concern, we use a cumulative
function to account for sampling biases in low access regions (?) in
illustrations in Figure 2.

Figure 2: (A) Cumulative density of accessibility and cumulative population
proportion. On the y-axis the dotted line represents the log of the cumulative
density of accessibility, and the triangle symbol line represents the log of
the cumulative density of population proportion. The x-axis represents the
accessibility score A. (B) Cumulative population proportion versus
accessibility score A for the 54 cities studied. $S(x)$, where
$S(x)=\frac{1}{1+e^{-k(x-0.5)}}$ where $k=10$, can be used as a benchmark for
accessibility within cities, where $50\%$ of the population has at least an
accessibility score A of 0.5. The dots and triangles represent equal sized
0.0625 km2 spatial units within and across all cities.

To understand this relationship better - the cumulative density (spatial
units) of accessibility (CDF($A$)) and the cumulative population density of
accessibility (CDF($p$)) - we present a ranking and a quadrant grid in Figure
3. Figure 3A represents a ranking of cities using $S(x)$ as a benchmark where
$50\%$ of the population has at least an accessibility score $A$ of $0.5$.
Cities are ranked according to the normalized mean absolute error (MAE)
between a city’s population curve and the benchmark function $S(x)$. In Figure
3B, we plot each city’s area under its (normalized) CDF($A$) and CDF($p$)
against each other. Each of the four quadrants of this grid represents a
unique scenario. Cities that fall in the top left quadrant (Q1) represent a
scenario where a large number of spatial units have high accessibility scores
but the majority of the population is concentrated in spatial units of lower
accessibility. In the particular case of Amsterdam in our data set, the city
witnesses very high levels of tourism every year. Most spatial units in the
city centre that have very high access, also see a huge crowd of foreign
visitors (?). Other spatial units in the city that are more residential, have
lower levels of relative access within the city, but much better absolute
levels of access when compared to other cities. This scenario is not observed
in other cities where tourism is not so prevalent, hence the scarcity of
points within this quadrant. Cities that fall in the top right quadrant (Q2)
represent a scenario where a large number of spatial units have low
accessibility scores and the majority of the population is concentrated in
spatial units of low accessibility. This scenario is very common for the
United States of America (USA) (for example, Los Angeles, Houston or Seattle)
which is entrenched in policies steering away from public transportation, and
a majority of the urban area is characterised by low-density suburban built
environment (?). Cities that fall in quadrant Q3 provide highly accessible
amenities to the most dense regions, with very limited pockets of less
accessible urban areas, but may be undervaluing areas with low density.
Quadrant Q4 illustrates cities where both low and high density spatial units
have higher levels of access, but the cities may be more sprawled with some
regions where access is low.

Figure 3: (A) Ranking of cities using $S(x)$ as a benchmark where $50\%$ of
the population has at least an accessibility score $A$ of $0.5$. Cities are
ranked according to the normalized mean absolute error (MAE) between a city’s
population curve and the benchmark function $S(x)$. (B) The four quadrants of
accessibility versus population densities. Q1 represents a scenario where a
large number of spatial units have high accessibility scores but the majority
of the population is concentrated in spatial units of lower accessibility. Q2
represents a scenario where a large number of spatial units have low
accessibility scores and the majority of the population is concentrated in
spatial units of low accessibility. Cities in Q3 provides highly accessible
amenities to the most dense regions, with very limited pockets of less
accessible urban areas. Quadrant Q4 has cities where both low and high density
spatial units have higher levels of access, but the cities may be more
sprawled with some regions where access is low.

The variation in accessibility by urban profiles

Above, we presented results on the static distribution of accessibility in any
given city without differentiating among the resident population. Cities in
the third quadrant of Figure 3B have a high median accessibility, indicating
that most of the population is also able to easily access urban infrastructure
in their vicinity. In comparison, many cities fall in quadrant $2$, where the
accessibility distribution is characterised by shorter tails and wider humps
(hence more inequalities prevalent among its residents - in Ref. (?), van Wee
et. al. discuss how a Gini index measure can be used to assess levels of
accessibility using density based distributions of accessibility). While this
analysis of accessibility as a resource is useful, it might prevent
recognition of structural inequalities in the distribution of this resource.
To provide appropriate support to decision-makers in addressing socioeconomic
inequalities in cities, we examine the relationship of accessibility
distributions with socioeconomic attributes of households within the spatial
units of a city using Chicago, IL (USA) as an illustrative example.

We cluster households using socioeconomic attributes to form urban profiles
where households _within_ a profile are similar to each other than _across_
profiles (see Section Methods for description of census data and clustering
methodology). We selected attributes that were largely available in the census
data across Canada and USA: ethnicity and minority status, income level,
marital status, household composition, language abilities, education level and
type, employment status, occupation type and commuting mode. A complete list
of demographic attributes is provided in the SI. By using consensus clustering
(?) on this set of attributes associated with Dissemination Areas (DA) (in
Canada) and Census Block Groups (CBG) (in USA) (see Section Methods for
modelling details), we identified a set of urban profiles for each city in our
original subset (in the Methods section we explain the choice for the subset
sample for further analysis). For instance, five urban profiles are identified
for the city of Chicago:

  * •

(1) Low income mixed (LIMix)

  * •

(2) Low income minority (LIM)

  * •

(3) Medium income white (MIW)

  * •

(4) High income white (HIW)

  * •

(5) Medium income white suburban (MIW-Suburban)

We observe that the resulting clusters vary between $3-5$ by city because of
each city’s unique socioeconomic distribution. The difference in the number of
profiles for various cities is also affected by the choice of the consensus
function used to evaluate the performance of clustering (see Methods for
clustering performance metrics). For example, for Chicago, IL (USA) based on
the k-modes consensus function, the most appropriate number of clusters is
equal to 5. Importantly, for some cities the difference in the metric values
is hardly distinguishable. In such cases we try to minimize the resulting
number of clusters while preserving subjective cluster interpretability.

Next, to establish a relationship between the accessibility score $A$ and
urban profiles, we interpolate each city’s clustered DAs or CBGs onto equally
sized spatial units. Although Chicago’s median accessibility of 0.51 makes it
generally more accessible than other cities, investigating accessibility
scores for each of Chicago’s urban profiles reveals more nuanced socioeconomic
differences. On average, urban areas inhabited by the most disadvantaged group
in Chicago are $27\%$ less accessible than those inhabited by the least
disadvantaged group. Figure 4A visually depicts the spatial differences in
accessibility for each urban profile, largely indicating the disparity in
accessibility between groups (2) LIM and (4) HIW, where a clear difference in
contrast is visible between the areas where the two groups live. Figure 4B
illustrates the distribution of accessibility across the urban profiles in
Chicago. We observe that in Chicago, urban profile (5) MIW-Suburban,
represents a small fraction ($10.7\%$) of spatial units mostly located in
suburban areas where low accessibility scores are likely reflective of
personal lifestyle choices rather than discriminatory outcomes of public
policy (?). We also observe that the (2) LIM urban profile has the least
median accessibility across the other groups (sans profile (5) MIW-Suburban).
The prevalent socioeconomic features of this profile are characterised by the
lowest levels of income, the highest percentage of individuals who identify as
visible minorities, the lowest number of individuals with university
education, and the highest rates of unemployment (Figure 4C). In contrast, the
group with the highest median accessibility - (4) HIW - is in stark contrast
with group (2) LIM, displaying diverging distributions for each socioeconomic
attribute (Figure 4C). If we break down accessibility by category of amenities
(Figure 4D), we observe that in Chicago, the highest degrees of inequality are
found in access to infrastructure related to food choices, entertainment,
active living, and health and well being. These results highlight areas for
intervention, where urban planners may focus equitable accessibility policies
in the city of Chicago. The results of other cities of the subset are reported
in the SI, and we can observe that in a majority of the subset of cities, low
income communities who also have a larger share of minorities, earn less and
are less educated, typically have lower access to urban amenities compared to
other communities in the region.

Figure 4: (A) Spatial representation of the five clustered social groups for
Chicago and the spatial distribution of accessibility for Chicago, where each
spatial unit (an equal sized square of 0.0625 km2) has its own accessibility
score. (B) The statistical distribution of accessibility for each of Chicago’s
clustered social groups. (C) Statistical distribution of normalized key
demographic attributes for each of Chicago’s clustered social groups. (D) A
parallel coordinates chart illustrating each social group’s median
accessibility scores dis-aggregated across amenity types.

Accessibility is inequitably distributed across urban profiles

We rank each urban profile based on their relative attribute values of income,
education, minority status, and employment census variables. In each city, we
choose the two groups at each extreme and classify them as either ”most
disadvantaged” (lowest income, lowest levels of education, highest proportion
of minorities, highest unemployment) or ”least disadvantaged” (highest income,
highest levels of education, lowest proportion of minorities, lowest
unemployment). In highly unlikely cases where the differences in socioeconomic
variables are negligible between two groups, we combine them together and
treat them as one. In each city, by comparing spatial units pertaining to
either group with their associated accessibility scores, we calculate the
probability density functions of each group’s respective accessibility. The
assessment of equity is determined by computing the relative change between
the two groups’ median accessibility as follows,

$\begin{split}A_{\Delta}(x,y)=\frac{A_{x}-A_{y}}{A_{y}},\end{split}$ (1)

where $A_{x}$ and $A_{y}$ are the respective median accessibility scores of
the “least disadvantaged” and “most disadvantaged” groups.

Figure 5 shows results from all $10$ North American cities (subset of this
work). We find that inequalities in access to urban infrastructure are present
in the majority ($6$ out of $10$) of cities we studied. The most disadvantaged
groups are structurally under-served by urban infrastructure as compared to
least disadvantaged groups. When accessibility is aggregated by category of
amenities (Figure 5), there is a gap in access to food, culture and
entertainment, active living and health infrastructure, which is all more
accessible in spatial units where the least vulnerable populations reside. The
subset analysis illustrates that in North American cities, public transit
options (represented by the mobility infrastructure) are scarcely distributed:
they are either very few (?) and difficult to access or have poorly evolved
through complex processes of institutions, government and corporate practices
that shape the choices of communities to be car-dependent (?).

Figure 5: Median accessibility scores for most advantaged and least advantaged
groups aggregated by city (A) and category (B) for $10$ North American cities

## Discussion

Our goal in this paper was to investigate the spatial relationship of
accessibility and urban inequalities. We found that the spatial distribution
of accessibility to infrastructure follows a universal log-normal paradigm,
indicating stratified urban communities (some with increasingly high access,
and others with lower levels of access). In accordance with Zipf’s laws of
cities (?), and considering the economic integration of urban regions (?),
some cities exhibit a widening inequality across social and spatial lines. Our
findings suggest that the distribution of amenities in cities plays a
significant role in how inequalities in access to urban infrastructure
manifest. The most socioeconomically disadvantaged groups are structurally
under-served by urban infrastructure as compared to least disadvantaged
groups. When dis-aggregating accessibility over the types of urban amenities,
there is an overall gap in access to food, culture and entertainment, active
living and health infrastructure, which is all more accessible in spatial
units where the least vulnerable profiles reside. The societal impacts of
these general inequality patterns in access to infrastructure can be serious.
For example, in a study of 11,599 Philadelphia households (?), the authors
report that households who reported very difficult access to fruits and
vegetables were more likely to also report food insecurity in their household.
In this context, poor access to food options in communities that are already
marginalized socioeconomically can contribute to perpetuate structural
inequalities among urban communities. While these findings are not surprising
and have been reported in multiple contexts and cases studies (?, ?, ?), a
framework to assess equity is an important step to understand the global state
of inequalities in accessibility from a spatial perspective. Through this
research we have unified previous social, empirical and theoretical work and
identified the general social groups that suffer from inequitable
accessibility to urban amenities.

Our research provides a holistic and reproducible framework for municipalities
worldwide to redress inequalities in access to infrastructure. Because of its
modular nature, the framework can be adapted to incorporate the multiple
values of stakeholders with regards to accessibility, and support in
identifying spatial regions and social groups suffering from inequalities in
access. Starting with global case studies of countries, regions or political
contexts, a lot of scholarly work can be reproduced or extended and
strengthened with our framework and access to open-source data. For example,
in Ref. (?), authors concluded that white and high-income communities in
Brazil have more access to jobs and education than black and poor communities,
irrespective of transport options considered. Thus, by combining measures of
active accessibility and dynamic measures of opportunities, such an approach
can understand the big picture better, and identify pockets of poorly
accessible neighbourhoods that also have lower access to job and other
opportunities across an urban region. Similarly, combining our insights with
work of scholars in Ref. (?), policymakers could derive targeted and equitable
measures for facilitating the distribution of amenities in neighbourhoods that
are most in need of such resources. Although, studying overall accessibility
levels do not support particular communities in understanding their needs and
contextual factors that perpetuate inequalities, our proposed approach can
provide an empirical foundation to narrow down on opportunities for decision-
makers to investigate further in remedying poor levels of access and
inequalities. When moving to a local case study of access to particular
infrastructure, this study can support scholars and policymakers in
identifying situation specific factors for their neighbourhoods. This can be
achieved by supplementing such a case study with other data sets that expand
on in-depth socioeconomic conditions of social groups over time, and how
active accessibility indicators have shaped in and around the region in
conjunction.

Although this work supports in identifying general levels of accessibility for
various urban communities, the results are bound to the definition of
accessibility. The accessibility score we define is based on the importance
attributed to each category encoded as a weight (same for all spatial units
and cities; see Methods). If the weights are changed drastically, the
distributions change quite considerably. Following the work of Maslow (?), we
agree that changing these weights drastically mean little to our understanding
of the basic needs of peoples. In case where decision-makers pander to
different politics where our needs change or must change considerably (and
thus the weights), the associated distributions of accessibility will paint a
different picture recognising the goals of policymakers as opposed to the
present levels of access that are afforded to communities. As our framework
design is modular and these weights can be changed, the associated
distributions of accessibility can help in understanding local measures of
access and contextualising them within the larger policy goals of the city
itself. We propose two avenues for future work in this regard. In one, more
practice-oriented work can be done to decipher the needs of communities
through public participation methods. These can be translated to weights that
are region specific and compared more broadly to city-wide levels of access.
In two, a theoretical approach can be used carry out a sensitivity analysis
(for instance, a Sobol Salteli sampling) on the weights to evaluate the
robustness of the [different] definition of accessibility. The outcome of such
an analysis; results of the changes in distributions of the accessibility
density plots can support in multiple policy goals where negotiations might be
necessary among decision-makers and stakeholders for further development.

Urban planning varies in practice across cities around the world. Diverse and
competing values, and regulatory policies around market instruments usually
govern how accessibility to amenities is shaped (?, ?). This is coupled with
more bottom-up processes of change within neighbourhoods (effects of local
economies, segregation and gentrification) that are shaped by the availability
(or lack thereof) of good quality infrastructure. While all of the subset
cities that we have studied have implemented policies targeted at improving
equity in accessibility (?), the presence of such a gap in accessibility
indicates that de facto processes of urban change and de jure policies
(whether discriminatory government actions of the past (?, ?), or utilitarian
policies and neoliberal market investments (?)) still perpetuate social
inequalities in today’s urban communities. Additionally, these results show
that present accessibility policies must be more ambitious, so as to reduce
the inequalities that exist in access to infrastructure to acceptable levels
and improve the minimum access affordability for multiple urban communities.

## Materials and Methods

### Accessibility Score

To compute accessibility, we use the amenity location point data (from here on
called Points of Interest (POI)) and the pedestrian street network data
collected from OpenStreetMap, where each node represents a street intersection
and each edge represents a pedestrian road segment. POIs do not warrant equal
importance for all people and contribute differently to addressing
inequalities (?). For example, improving access to a pharmacy or library might
be more important to a social group than access to a public transport station.
Maslow’s hierarchy of needs suggests, that in general, humans have three
important needs in their living environment: Physiological, Safety and
Security, and Social needs (?). This suggests that similar POIs can be
classified into categories for ease of analysis and to reflect more broadly on
the concept of accessibility and inequalities. So, to capture the essential
aspects of urban accessibility, we classify each POI into one of seven
categories of accessibility: Mobility, Active living, Entertainment, Food
Choices, Community Space, Education, Health and Well Being (Supplementary Note
1). We choose these categories to reflect on the needs of peoples (as
specified by Maslow) and broader attributes of functional and livable urban
spaces: transportation, public health, food security, cultural capital, and
social cohesion (?, ?). We (the 3 co-authors) carried out a subjective
assignment of the POI to the seven categories. Then we deliberated on where we
differed, and found resolution within the context of needs and livable urban
spaces. Next, each point, for each category, is assigned to the nearest edge
on the pedestrian network. For each node on the network, we calculate the
routing distance (weighted shortest path along the pedestrian network) to the
nearest POI. Each node on the network is then assigned a value representing
the distance in meters to the nearest amenity point. This process is repeated
for each category of amenities, resulting in each node being assigned seven
accessibility values. Previous studies on quantifying accessibility in cities
have focused on measuring the Euclidean distance to services (?). However, the
ease with which residents can access services in a city is limited by urban
form and land-use and the routing distance is a better proxy for the
accessibility (?).

With the use of Python’s open source _OSMnx_ library (?), we filter
OpenStreetMap according to different keys (“public transport”, “leisure”,
“amenity”). We create a geospatial point data set, consisting of $50$ types of
urban amenities, of all the POIs considered in each city. In total we collect
$1,678,000$ POIs across all $54$ cities. SI reports the number of points
collected for each city.

Next, we use _OSMnx_ to download street networks for the pedestrian street
infrastructure of each city. Using the street network data, we interpolate
nodes within identical 0.0625 km2 spatial units in a grid matching those
resulting from the analysis of social groups. By averaging distances from the
nodes that fall within each spatial unit to nearest amenities of each category
(that may or may not be in the same spatial unit), we compute the average
walking distance within every spatial unit of a city to each of the seven
categories of basic amenities. Finally, we weigh and aggregate average walking
distances into a walking distance measure $D$ using the following equation:

$\begin{split}D_{i}=\log(\sum\limits_{c}w^{c}d_{i}^{c}),\end{split}$ (2)

where $d_{i}^{c}$ is the average walking distance from nodes that fall within
spatial unit i to the nearest amenity of a given category c, and w (category-
wise constant across all spatial units and cities) is the weight attributed to
that category. See SI for the values of weights to reproduce this analysis.
The use of a logarithmic function captures the outliers in the measurement and
makes comparison easier within and across cities. Using a min-max scaler
function, we normalize measure $D$ within each city so we can normatively rank
walking distances of every spatial unit relative to each city:

$\begin{split}\widetilde{D_{i}}=\frac{D_{i}-min(D_{i})}{max(D_{i})-min(D_{i})}.\end{split}$
(3)

Finally, we subtract $\widetilde{D\textsubscript{i}}$ from $1$ to obtain an
accessibility score $A$, where large values indicate high accessibility and
low values indicate low accessibility:

$\begin{split}A_{i}=1-\widetilde{D_{i}}\end{split}$ (4)

### Data-driven Framework

To understand the nature of inequalities that citizens face in accessibility
to urban infrastructure, we introduce a data-driven framework that facilitates
comparison of accessibility scores of groups of individuals that share similar
demographic and socioeconomic attributes (e.g. age, ethnicity, income,
employment, mobility) within and across cities worldwide (Figure 6). This
framework is based on three types of data sets (see Section Data Description).
First, we use census data from governmental agencies to identify which
population groups are predominant within a certain area - these are called
social groups (see Section Data Description for detailed information on census
data). We use consensus clustering (see Section Clustering Method) to identify
social groups using variables of interest. In parallel, we perform a network
analysis on the street networks and urban amenities data to quantify and map
accessibility to urban infrastructure at the block level. Finally, we measure
inequalities by comparing the distribution of accessibility across social
groups that were identified through consensus clustering. In the spirit of
contributing to open-source data and research (?), our framework is fully
automated and documented as open-source code on GitHub (?).

We apply this framework to $54$ cities across _six_ continents as reported in
the Tables in the SI. We measure accessibility for all $54$ cities and examine
the variability in its distribution for the city population on a subset of
those cities: New York City (NY), Chicago (IL), San Francisco (CA), Los
Angeles (CA), Houston (TX), Seattle (WA), Miami (FL), Toronto (ON), Vancouver
(BC), and Montreal (QC). Within this subset, the municipal authorities of each
city have made strong policy efforts to provide equitable accessibility to
urban services within their municipal jurisdiction (?). In addition, open-
source data programs run by the municipal governments of these cities provide
aggregated demographic and socioeconomic variables for the complete
population.

Figure 6: Framework for Evaluating Equity in Accessibility to Urban
Infrastructure.

#### Data Description

Census Data. To characterize population demographics, we use $2016$
demographic data from Statistics Canada (?) at the Dissemination Area (DA)
level, and 2017 demographic data estimates from the United States Census at
the Census Block Group (CBG) (?) level. In order to capture a full spectrum of
demographic attributes, we consider $46$ census variables relating to
ethnicity and minority status, income level, marital status, household
composition, language abilities, education level and type, employment status,
occupation type and commuting mode. Several scholars agree that many of these
socioeconomic variables are related to factors explaining inequalities in
cities (?), although the evidence is scattered in the literature through case
studies specific to cities or types of infrastructure.

Streets and Amenities. To characterize street networks and urban amenities, we
use OpenStreetMap (?) to collect data on pedestrian infrastructure and
geographically allocated places of interest (POI): hospitals, schools,
supermarkets, restaurants, schools, etc. POIs are returned by the
OpenStreetMap API as amenity names and their associated geographical location
(i.e. longitude and latitude). Pedestrian infrastructure networks are returned
by the OpenStreetMap API as networks of nodes and edges, where each node
represents a street intersection and each edge represents a segment of road
with walkable or bikable features. Next, we group POI types into a set of
different categories (see SI for a complete list of amenities and their
associated category).

This source is inconsistent in terms of data completeness and quality. The
street network can be incomplete in several rural and developing parts of the
world. The number of amenities in OSM can be twice as high as the real number
(for instance, the number of POIs categorized as schools for Helsinki is
around 250 whereas the real (from the city’s administrative data base) is
100). In some cases, this happens because the name of buildings occur twice or
thrice, like school building A, and B. In our study, we are not interested in
the number of those amenities, rather, the location is important. Further, the
cities we have selected have a reliable amount of data ($>80\%$) available
from OpenStreetMap, and in most cases, the data is complete even in the
developing world (?).

Population Grids Data on population density for every city is retrieved from
the European Commission’s 2015 Global Human Settlement Layer (GHSL) (?). This
data is retrieved in the form of a grid of 250m by 250m squares (0.0625 km2
spatial units exactly the same size as units used for accessibility analysis)
and their associated population density values covering the entire world.

Year mismatch in the collection of data sets The data sets we have used in
this analysis have been collected and catalogued across different years. The
census data is from the year 2016 (Canada) and 2017 (US). The street networks
were extracted in the year 2019. The GHSL grid data of population density is
from 2015. This does not cause any problems for comparison as we do not expect
population densities to change drastically within a couple of years. Further,
we are not making any inferences about certain people but a group as a whole,
and do not expect neighbourhoods to gentrify within a year in any city in the
world.

### Clustering Method

Consensus clustering is an unsupervised machine learning method that is widely
used in the fields of molecular biology and complex networks (?, ?, ?). While
it has recently found useful applications in the field of social network
analysis (?, ?), there is not much evidence for its use in urban science (?,
?). The method combines and evaluates results of several clustering algorithms
to find the most robust set of labels for clusters (?) in the given data set.
The final decision about the labels is made based on a consensus function
(e.g. Majority Voting (?)). In this work, we use the k-modes consensus
function (?):

$\begin{split}d(x,y)=\sum\limits_{i=1}^{m}(1-\delta(x_{i},y_{i})),\end{split}$
(5)

where x and y are two categorical objects (spatial units) with m categorical
attributes (demographic and socioeconomic variables), and the total mismatches
of the corresponding attribute categories of the two objects represent the
dissimilarity measure between x and y (?).

Authors of Ref. (?) showed that such an approach outperforms other consensus
functions and has low computational complexity. However, since the true labels
are unknown, we have not evaluated results of other consensus functions versus
k-modes. In future endeavours, it might be interesting to validate the
clusters with expert knowledge of the urban region, and compare that with
other consensus functions.

We manually investigated over 16,000 socioeconomic variables for the US and
over 4,000 for Canada and selected 44 variables of two categories of interest:
socioeconomic (age, ethnicity, income, employment) and housing-related (type
of house ownership, average housing price). The number of observations (the
sample) differ for each of the cities. All features are numerical with most of
them reported as percentages. The resulting data set has outliers. There are
areas in cities where, for example, population density is significantly
smaller than in the rest of the blocks. To include all areas in the analysis
appropriately, we reduce the influence of outliers through a technique called
winsorizing (?): limiting the number of extreme values in the data set. We set
up the limits as 0.1 percentile and 0.9 percentile respectively, hence
subjecting the data to 80% winsorization. In the final pre-processing step, we
standardize all features by removing the mean and scaling the observations to
unit variance.

To implement consensus clustering in our analysis we use the diceR library for
the R programming language, which uses an ensemble clustering framework to
perform cluster analysis (?). We iterate over 5 clustering algorithms, namely
K-means, K-medoids, Hierarchical clustering, Gaussian mixture model and
HDBSCAN. diceR also requires to specify distance functions. We select 7
distance functions aiming to cover all potential issues with the data:
Euclidean, Maxmium, Manhattan, Canberra, Binary, Minkowski and Spearman. The
algorithm runs 10 replications and gives us cluster labels based on k-modes
consensus function.

### Spatial Units

In this study, each spatial unit is described by a 0.0625 km2 square
originating from the GHSL population grid (?). The main spatial unit is a
square that does not delineate the spatial context on the ground (data
observation or public administration of space as differently sized and
populated administrative districts). We use square grids to reduce sampling
bias from edge effects (?) and minimize the Modifiable Areal Unit Error (MAUP)
(?). Although this framing does resolve the MAUP partially, it does not
represent a spatial unit that is directly linked to the problem at hand, and
thus introduces MAUP on its own. Like any spatial unit that brings in a degree
of MAUP, squares and hexagons have the ability to partly address the size
problem. If the data is also collected keeping in mind the framed boundaries
of equally-spaced grid-cells, a better resolution could be achieved. However,
this is not the case with population and accessibility variables, and
interpolation of these data sets is bound to bring in some error. For each
city, the GHSL grid is clipped to the city’s administrative boundaries and is
used to calculate the accessibility score (A) within each of its squares. For
each of the 10 cities in the subset analysis of socioeconomic variables, this
same grid is used to assign cluster values obtained from clustering using
census geometries. This process involves (a) checking whether a given square
spatially intersects with a given census geometry, (b) assigning this census
geometry’s cluster value to the given square if this condition is met. Note
that our methodology here is limiting in nature. Each census geometry is
assigned one cluster label from the consensus clustering algorithm. The square
spatial units that fall entirely within the census geometries get one clearly
defined label attributed to it. However, for squares that fall at the boundary
of census geometries and the boundary of two clusters ($<2\%$ in the entire
data set), their assignment to a cluster is based only on the order in which
they are being checked.

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