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

\begin{align} w_0 = a, \quad w_1 = a+d, \quad w_2 = a+2d, \quad \ldots, \quad w_k = a+kd. \end{align}


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

\begin{align} A(x) := \max \left\{ |S| \mid S \subseteq \{1, \ldots, x\}, S \text{ an } \mathcal{A}\text{-set} \right\} \end{align}

and define the function $a(x)$ by

\begin{align} a(x) := \frac{A(x)}{x}. \end{align}

Exponential Sum, S

For notational ease, we first define the function

\begin{align} e(\theta) = e^{2 \pi i \theta}. \end{align}

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:

\begin{align} S := \sum_{k=1}^U e(\alpha \cdot u_k). \end{align}

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:

\begin{align} \alpha = \frac{h}{q} + \beta; \end{align}

and they satisfy the following three conditions:

\begin{equation} (h,q) = 1 \end{equation}
\begin{align} q \leq \sqrt{M} \end{align}
\begin{align} q | \beta | \leq \frac{1}{\sqrt{M}} \end{align}

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:

\begin{align} S' := \frac{a(m)}{q} \cdot \left( \sum_{r=1}^q e\left( \frac{r \cdot h}{q} \right) \right) \cdot \left( \sum_{n=1}^M e(\beta \cdot n) \right) \end{align}

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

\begin{align} f_1(\alpha) = \sum_{k=1}^U e(\alpha u_k) \end{align}
\begin{align} f_2(\alpha) = \sum_{k=1}^V e(\alpha v_k) \end{align}
\begin{align} F_1(\alpha) = a(m) \sum_{n=1}^{2N} e(\alpha n) \end{align}
\begin{align} F_2(\alpha) = a(m) \sum_{n=1}^N e(\alpha n). \end{align}

Norm $||\alpha||$

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

\begin{align} \| \alpha \| \in \left[ 0, \frac{1}{2} \right]. \end{align}

Delta, $\delta$

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

\begin{align} \delta = \frac{1}{N \eta}. \end{align}

x and b(x), $x, b(x)$

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

\begin{equation} m = 2^{4^x} \end{equation}


\begin{align} b(x) = a(m) = a \left( 2^{4^x} \right) \end{align}