This page contains definitions either given in Roth's paper or pertinent to it.

### Arithmetic Progression

A set of numbers $w_0, w_1, \ldots, w_k$ are said to be an **arithmetic progression** is there exists an $a$ and $d$ such that

### A-Set

A subset $\{ u_1, u_2, \ldots, u_k \}$ of the natural numbers $\mathbb{N}$ is called an **$\mathcal{A}$-set** if no three elements form an arithmetic progression.

### Functions A(x) and a(x)

We define the function $A(x)$ by

(2)and define the function $a(x)$ by

(3)### Exponential Sum, S

For notational ease, we first define the function

(4)We then denote by $S$ the following exponential sum, defined on a set $\{u_1, \ldots u_U\}$ where $\alpha$ is an arbitrary, but predefined, real number:

(5)In our case, we will usually take $\{u_1, \ldots, u_U\}$ to be a maximal $\mathcal{A}$-set in $\{1, 2, \ldots, M\}$.

### Dirichlet Constants

For a given real number $\alpha$ we define the constants $h$, $q$ and $\beta$ by the Dirichlet Approximation Theorem. That is, so that:

(6)and they satisfy the following three conditions:

(7)### Exponential Sum, S'

Let $m < M$ and denote by $S'$ the following sum, where $h$, $q$ and $\beta$ are found by the Dirichlet Approximation Theorem:

(10)where $\{ u_1, u_2, \ldots, u_U\}$ is a fixed $\mathcal{A}$-set in $\{1, \ldots, M\}$.

### Constants for the Hardy-Littlewood Method

We define $m$ to be even, so that $m^4$ is also even. We therefore define $N$ such that $2N = m^4$. We fix $\{u_1, \ldots, u_U\}$ a maximal $\mathcal{A}$-set from $\{1, 2, \ldots, 2N\}$. We define the $v_i$ to be such that $2v_1, 2v_2, \dots 2v_V$ are the even $u_i$.

### Functions for the Hardy-Littlewood Method

We define the following functions for use in the adapted Hardy-Littlewood Method

(11)### Norm $||\alpha||$

For any real $\alpha$ we define the quantity $\|\alpha\|$ to be the distance from $\alpha$ to the nearest integer. Hence

(15)### Delta, $\delta$

For $0 < \eta < \alpha < 1 - \eta$ we write

(16)### x and b(x), $x, b(x)$

We define $x$ and $b(x)$ by

(17)and

(18)