On K. F. Roth’s ‘On Certain Sets of Integers’

The aim of this paper is to inspect [6] and work through to make every step transparent and clear. Main Result From [6] In [6] Roth looked at three term arithmetic sequences in certain sets of integers. He considered a function, called A(x), that was the maximum size of a subset of {1, 2, . . . , x} that avoided three term arithmetic sequences. Having already proven the main result: A(x) x −→ 0 as x −→∞ Roth used the paper to proved a tighter aymptotic and so proved the following: Theorem 1 (Roth’s Theorem). A(x) x = O ( 1 log log x ) 1 (1) The standard O notation is that f(y) = O(g(y)) if and only if there exists an absolute constant M such that for all sufficiently large y, |f(y)| ≤M · |g(y)|.