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
(1)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)