Key Definitions

This page contains explanations pertaining to the "Key Definitions" section of Roth's paper.

Alternative Definition of Arithmetic Sequence

A set is an $\mathcal{A}$-set if and only if the only solution to $u_j + u_l = 2u_k$ is when $j=k=l$.

If a set $A = \{u_1, u_2, \ldots\}$ contained an arithmetic progression $u_j, u_k, u_l$ then there would exist positive integers $a$ and $d$ such that

(1)
\begin{align} u_j=a; \qquad u_k = a+d; \qquad u_l=a+2d; \end{align}

so that $u_j+u_l = 2u_k$. On the other hand, if there existed a non-trivial solution to $u_j + u_l = 2u_k$ then setting $a$ to be $u_j$ and $d$ to be $u_k - u_j$ exhibits an arithmetic progression.